Is Dialetheism Self-Coherent?
Ederson Safra Melo
Jonas R B ArenhartSouth American Journal of Logic
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Dialetheism is the view according to which some contradictions are true. The main motivation for such a view comes from the well-known Liar paradox. The dialetheist simply takes the Liar argument as legitimate; by embracing a true contradiction as acceptable, the problem is thought to be solved. We shall argue that there are difficulties in explaining what dialetheism really amounts to. The first main obstacle comes from the fact that there are distinct versions of the Liar. According to the dialetheist, they do not mean the same thing, and as we shall argue, there is no unified dialetheist treatment for them. The second main problem comes from the very idea of true contradictions and truth value gluts. There are two distinct possible understanding for those concepts, and they do point to conflicting understandings of dialetheism. It is not even clear how to conciliate the dialetheists' guiding intuitions with those problems, so that after all, it seems that dialetheism may end up being incoherent.
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❙♦✉t❤ ❆♠❡r✐❝❛♥ ❏♦✉r♥❛❧ ♦❢ ▲♦❣✐❝
❱♦❧✳ ✸✱ ♥✳ ✶✱ ♣♣✳ ✽✼✕✶✵✾✱ ✷✵✶✼
■❙❙◆✿ ✷✹✹✻✲✻✼✶✾
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
❏♦♥❛s ❘✳ ❇❡❝❦❡r ❆r❡♥❤❛rt ❛♥❞ ❊❞❡rs♦♥ ❙❛❢r❛ ▼❡❧♦
■♥ ❛ s❡♥s❡✱ t❤❡r❡❢♦r❡ ♠② ♣♦s✐t✐♦♥
✐s q✉✐t❡ s❡❧❢✲❝♦♥s✐st❡♥t✱ t❤♦✉❣❤
t❤✐s ✐s ❤❛r❞❧② ❛ ❤❛♣♣② ✇❛② ♦❢
❡①♣r❡ss✐♥❣ t❤❡ ♠❛tt❡r✦ ❙♦ ❧❡t ✉s
s❛② t❤❛t ✐t ✐s s❡❧❢✲❝♦❤❡r❡♥t✳
●✳ Pr✐❡st✱
■♥ ❈♦♥tr❛❞✐❝t✐♦♥✱ ♣✳ ✼✷✳
❆❜str❛❝t
❉✐❛❧❡t❤❡✐s♠ ✐s t❤❡ ✈✐❡✇ ❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ s♦♠❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛r❡
tr✉❡✳ ❚❤❡ ♠❛✐♥ ♠♦t✐✈❛t✐♦♥ ❢♦r s✉❝❤ ❛ ✈✐❡✇ ❝♦♠❡s ❢r♦♠ t❤❡ ✇❡❧❧✲❦♥♦✇♥ ▲✐❛r
♣❛r❛❞♦①✳ ❚❤❡ ❞✐❛❧❡t❤❡✐st s✐♠♣❧② t❛❦❡s t❤❡ ▲✐❛r ❛r❣✉♠❡♥t ❛s ❧❡❣✐t✐♠❛t❡❀ ❜②
❡♠❜r❛❝✐♥❣ ❛ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥ ❛s ❛❝❝❡♣t❛❜❧❡✱ t❤❡ ♣r♦❜❧❡♠ ✐s t❤♦✉❣❤t t♦
❜❡ ✏s♦❧✈❡❞✑✳ ❲❡ s❤❛❧❧ ❛r❣✉❡ t❤❛t t❤❡r❡ ❛r❡ ❞✐✣❝✉❧t✐❡s ✐♥ ❡①♣❧❛✐♥✐♥❣ ✇❤❛t
❞✐❛❧❡t❤❡✐s♠ r❡❛❧❧② ❛♠♦✉♥ts t♦✳ ❚❤❡ ✜rst ♠❛✐♥ ♦❜st❛❝❧❡ ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t
t❤❛t t❤❡r❡ ❛r❡ ❞✐st✐♥❝t ✈❡rs✐♦♥s ♦❢ t❤❡ ▲✐❛r✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❞✐❛❧❡t❤❡✐st✱
t❤❡② ❞♦ ♥♦t ♠❡❛♥ t❤❡ s❛♠❡ t❤✐♥❣✱ ❛♥❞ ❛s ✇❡ s❤❛❧❧ ❛r❣✉❡✱ t❤❡r❡ ✐s ♥♦
✉♥✐✜❡❞ ❞✐❛❧❡t❤❡✐st tr❡❛t♠❡♥t ❢♦r t❤❡♠✳ ❚❤❡ s❡❝♦♥❞ ♠❛✐♥ ♣r♦❜❧❡♠ ❝♦♠❡s
❢r♦♠ t❤❡ ✈❡r② ✐❞❡❛ ♦❢ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛♥❞ tr✉t❤ ✈❛❧✉❡ ❣❧✉ts✳ ❚❤❡r❡ ❛r❡
t✇♦ ❞✐st✐♥❝t ♣♦ss✐❜❧❡ ✉♥❞❡rst❛♥❞✐♥❣ ❢♦r t❤♦s❡ ❝♦♥❝❡♣ts✱ ❛♥❞ t❤❡② ❞♦ ♣♦✐♥t
t♦ ❝♦♥✢✐❝t✐♥❣ ✉♥❞❡rst❛♥❞✐♥❣s ♦❢ ❞✐❛❧❡t❤❡✐s♠✳ ■t ✐s ♥♦t ❡✈❡♥ ❝❧❡❛r ❤♦✇ t♦
❝♦♥❝✐❧✐❛t❡ t❤❡ ❞✐❛❧❡t❤❡✐sts✬ ❣✉✐❞✐♥❣ ✐♥t✉✐t✐♦♥s ✇✐t❤ t❤♦s❡ ♣r♦❜❧❡♠s✱ s♦ t❤❛t
❛❢t❡r ❛❧❧✱ ✐t s❡❡♠s t❤❛t ❞✐❛❧❡t❤❡✐s♠ ♠❛② ❡♥❞ ✉♣ ❜❡✐♥❣ ✐♥❝♦❤❡r❡♥t✳
❑❡②✇♦r❞s✿ ❞✐❛❧❡t❤❡✐s♠✱ ❝♦♥tr❛❞✐❝t✐♦♥✱ ▲✐❛r ♣❛r❛❞♦①✱ tr✉t❤✲✈❛❧✉❡ ❣❧✉ts✳
✶
■♥tr♦❞✉❝t✐♦♥
❉✐❛❧❡t❤❡✐s♠ ♠❛② ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❛s t❤❡ t❤❡s✐s ❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ s♦♠❡ ❝♦♥✲
tr❛❞✐❝t✐♦♥s ❛r❡ tr✉❡✱ t❤❛t ✐s✱ s♦♠❡ ♣❛✐rs ♦❢ s❡♥t❡♥❝❡s ♦❢ t❤❡ ❢♦r♠
α
❛♥❞
¬α
❛r❡
❜♦t❤ tr✉❡✱ ✇❤❡r❡ ✏ ¬✑ ✐s t❤❡ ♥❡❣❛t✐♦♥ s✐❣♥❀ t❤❡s❡ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛r❡ ❝❛❧❧❡❞
✽✽
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
❞✐❛❧❡t❤❡✐❛s✳
●r❛❤❛♠ Pr✐❡st ✐s t❤❡ ❝❤❛♠♣✐♦♥ ♦❢ ❞✐❛❧❡t❤❡✐s♠✱ ❛♥❞ ✐♥ t❤✐s ♣❛♣❡r
✇❡ s❤❛❧❧ ❢♦❝✉s ♦♥ ❤✐s ✈❡rs✐♦♥ ♦❢ ❞✐❛❧❡t❤❡✐s♠✱ ❛❧t❤♦✉❣❤ t❤❡ ✈✐❡✇ ✐s ❝✉rr❡♥t❧② ❡s✲
♣♦✉s❡❞ ❜② ♦t❤❡rs ❛s ✇❡❧❧✱ ✇✐t❤ ✈❛r②✐♥❣ ❞❡❣r❡❡s ♦❢ ✐♥✈♦❧✈❡♠❡♥t ✭s❡❡ ❇❡❛❧❧ ❬✷❪
❢♦r ❛♥ ♦✈❡r✈✐❡✇❀ Pr✐❡st ❬✾❪ ❢♦r t❤❡ ❝❧❛ss✐❝❛❧ st❛t❡♠❡♥t ❛♥❞ ❞❡❢❡♥s❡ ♦❢ t❤❡ ✈✐❡✇✮✳
❲✐t❤ t❤❡ ♣r❡✈✐♦✉s ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥s ❛♥❞ ❞✐❛❧❡t❤❡✐s♠✱ ✐t ✐s s❛✐❞✱
s♦♠❡ s❡♥t❡♥❝❡s ✭t❤❡ ❞✐❛❧❡t❤❡✐❛s✮ ❛r❡ ❜♦t❤ tr✉❡ ❛♥❞ ❢❛❧s❡❀ t❤❛t ✐s✱ t❤❡② ✐♥❝❛r♥❛t❡
tr✉t❤ ✈❛❧✉❡ ❣❧✉ts✳ ❙♦✱ t❤❡ ♥♦t✐♦♥s ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❞✐❛❧❡t❤❡✐❛s✱ ❛♥❞ ❣❧✉ts ❛r❡
❝❧♦s❡❧② r❡❧❛t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞✐❛❧❡t❤❡✐st✳
Pr✐❡st ❛❞✈❛♥❝❡s t❤♦s❡ ❝♦♥❝❡♣ts ✐♥ ♥✉♠❡r♦✉s ♣❧❛❝❡s✳ ❋♦r ✐♥st❛♥❝❡✿
❉✐❛❧❡t❤❡✐s♠ ✐s t❤❡ ✈✐❡✇ t❤❛t s♦♠❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛r❡ tr✉❡✿ t❤❡r❡ ❛r❡
s❡♥t❡♥❝❡s ✭st❛t❡♠❡♥ts✱ ♣r♦♣♦s✐t✐♦♥s✱ ♦r ✇❤❛t❡✈❡r ♦♥❡ t❛❦❡s tr✉t❤✲
❜❡❛r❡rs t♦ ❜❡✮✱
α
t❤❛t
α✱
s✉❝❤ t❤❛t ❜♦t❤
α
❛♥❞
¬α
❛r❡ tr✉❡✱ t❤❛t ✐s✱ s✉❝❤
✐s ❜♦t❤ tr✉❡ ❛♥❞ ❢❛❧s❡ ✭Pr✐❡st ❬✶✵✱ ♣✳✶❪✮✳
❚❤❡ r❡❧❛t✐♦♥ ♦❢ ❞✐❛❧❡t❤❡✐s♠ ✇✐t❤ ❝♦♥tr❛❞✐❝t✐♦♥s s❤♦✉❧❞ ❜❡ ❝❧❡❛r ❢r♦♠ t❤❛t
q✉♦t❡ ✭s❡❡ ❛❧s♦ t❤❡ ♦♣❡♥✐♥❣ ♣❛r❛❣r❛♣❤ ✐♥ Pr✐❡st ❛♥❞ ❇❡rt♦ ❬✶✶❪✮✳ ❚♦ ❜r✐♥❣ ❣❧✉ts
t♦ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ st❛❣❡ t♦♦✱ Pr✐❡st s❛②s ❤✐s ♣♦s✐t✐♦♥ ✐s ♦♥❡
(. . . )
❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ ♣❛r❛❞♦①✐❝❛❧ s❡♥t❡♥❝❡s ❛r❡ ❜♦t❤ tr✉❡ ❛♥❞
❢❛❧s❡ ✖ ❛s ✐t ✐s s♦♠❡t✐♠❡s ♣✉t✱ t❤❡② ❛r❡ s❡♠❛♥t✐❝ ❣❧✉ts ✭❞✐❛❧❡t❤❡✐❛s✮✳
✭Pr✐❡st ❬✼✱ ♣✳✺✼❪✮
❚❤♦s❡ ❞❡✜♥✐t✐♦♥s ♦r ❝❧❛r✐✜❝❛t✐♦♥s ❛♣♣❡❛r ✐♥ ♣r❛❝t✐❝❛❧❧② ❡✈❡r② r❡❢❡r❡♥❝❡ ❛❜♦✉t
❞✐❛❧❡t❤❡✐s♠✳ ■♥ ♦r❞❡r t♦ s✉♠♠❛r✐③❡ t❤❡ ♠❛✐♥ ✐❞❡❛s ❛♥❞ ❢♦r ❢✉t✉r❡ r❡❢❡r❡♥❝❡✱ ✇❡
s❤❛❧❧ ❛❞♦♣t t❤❡ ❢♦❧❧♦✇✐♥❣ ❛s ♦✉r ♦✣❝✐❛❧ ❞❡✜♥✐t✐♦♥s ♦❢ t❤♦s❡ t❡r♠s✿
❈♦♥tr❛❞✐❝t♦r✐❡s✿ ❛ ♣❛✐r ♦❢ s❡♥t❡♥❝❡s ♦❢ t❤❡ ❢♦r♠
α
❛♥❞
¬α✱
✇❤❡r❡ ✏ ¬✑ ✐s
t❤❡ ♥❡❣❛t✐♦♥ s✐❣♥ ✭♦❜✈✐♦✉s❧②✱ t❤❡r❡ ✐s ❞✐s♣✉t❡ ♦♥ ✇❤❛t ♥❡❣❛t✐♦♥ ✐s ❛♥❞
❤♦✇ ❞♦❡s ✐t ❜❡❤❛✈❡❀ ✇❡ s❤❛❧❧ ♥♦t ❞✐s❝✉ss ✐t ❤❡r❡✱ ❜✉t r❛t❤❡r ❛ss✉♠❡ t❤❛t
♥❡❣❛t✐♦♥ ❜❡❤❛✈❡s ❛s t❤❡ ❞✐❛❧❡t❤❡✐sts s❡❡♠ t♦ ❛❣r❡❡✮✳
❈♦♥tr❛❞✐❝t✐♦♥✿ t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❝♦♥tr❛❞✐❝t♦r✐❡s✱ t❤❛t ✐s✱ ❛ s❡♥t❡♥❝❡ ♦❢ t❤❡
❢♦r♠
α ∧ ¬α✳✶
❉✐❛❧❡t❤❡✐❛✿ ❛ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥✱ t❤❛t ✐s✱ ❛ s❡♥t❡♥❝❡
❛♥❞
¬α
α ∧ ¬α
s✉❝❤ t❤❛t ❜♦t❤
α
❛r❡ tr✉❡ ✭♥♦t✐❝❡ t❤❛t ✐❞❡❛❧❧② ♥♦t ❡✈❡r② ❝♦♥tr❛❞✐❝t✐♦♥ ✐s tr✉❡✱ s♦✱
♥♦t ❡✈❡r② ❝♦♥tr❛❞✐❝t✐♦♥ ✐s ❛ ❞✐❛❧❡t❤❡✐❛❀ ♠♦r❡ ♦♥ t❤✐s ❧❛t❡r✮✳
❚r✉t❤ ✈❛❧✉❡ ❣❧✉t✿ ❛ tr✉t❤ ✈❛❧✉❡❀ ✇❤❡♥ ❛ s❡♥t❡♥❝❡
✈❛❧✉❡s✱ t❤❛t ♠❡❛♥s t❤❛t
✶
α
α r❡❝❡✐✈❡s ❛ ❣❧✉t ❛s ✐ts tr✉t❤
✐s ❜♦t❤ tr✉❡ ❛♥❞ ❢❛❧s❡✳
●✐✈❡♥ t❤❛t ❝♦♥❥✉♥❝t✐♦♥ ❜❡❤❛✈❡s ❛s ❡①♣❡❝t❡❞✱ ✇❡ ♠❛② ❣♦ ❢r♦♠ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ t♦ ❛ ♣❛✐r ♦❢
❝♦♥tr❛❞✐❝t♦r✐❡s ❜② s❡♣❛r❛t✐♥❣ t❤❡♠✱ ❛♥❞ ❢r♦♠ ❝♦♥tr❛❞✐❝t♦r✐❡s t♦ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ❜② ✐♥tr♦❞✉❝✐♥❣
t❤❡ ❝♦♥❥✉♥❝t✐♦♥✳
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✽✾
◆♦t✐❝❡ t❤❛t ✇❤✐❧❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥ ✐s ❢r❛♠❡❞ ✐♥ t❡r♠s ♦❢ t❤❡
s②♥t❛① ♦❢ ❛ ❧❛♥❣✉❛❣❡ ✭✐t r❡q✉✐r❡s ♦♥❧② t❤❛t ❛ ❝❡rt❛✐♥ ❢♦r♠✉❧❛ ❤❛s ❛ ❝❡rt❛✐♥ ❢♦r♠✮
❛♥❞ ❞❡♣❡♥❞s ♦♥ t❤❡ ♥♦t✐♦♥ ♦❢ ♥❡❣❛t✐♦♥✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❞✐❛❧❡t❤❡✐❛ ✐♥tr♦❞✉❝❡s
❛ s❡♠❛♥t✐❝ ❝♦♠♣♦♥❡♥t ✭✐ts tr✉t❤ ✈❛❧✉❡✮✱ ✇❤✐❧❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❣❧✉ts ✐s ❝❧❡❛r❧②
❢r❛♠❡❞ ✐♥ s❡♠❛♥t✐❝ t❡r♠s✱ ❛♥❞ ✐♥ ❢❛❝t✱ s❡❡♠s t♦ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♥❡❣❛t✐♦♥
✭♠♦r❡ ♦♥ t❤✐s ❧❛t❡r✮✳
❚❤❡ t②♣✐❝❛❧ ❞✐❛❧❡t❤❡✐st ✭♠❛✐♥❧② Pr✐❡st✱ ❜✉t ♦t❤❡rs ❛s
✇❡❧❧✮ ❝♦♥✢❛t❡s t❤❡ t❤r❡❡ ❝♦♥❝❡♣ts❀ t❤❡② ❛r❡ ❛❧❧ r❡q✉✐r❡❞ ❢♦r ❛ t②♣✐❝❛❧ ❞✐❛❧❡t❤❡✐st
❛❝❝♦✉♥t ♦❢ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s✳ ❚❤❡ ❢♦❝✉s ♦❢ t❤✐s ♣❛♣❡r ✇✐❧❧ ❜❡ t♦ ❡①♣❧♦r❡ t❤❡
r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤♦s❡ ❝♦♥❝❡♣ts ❛♥❞ ❝❤❛❧❧❡♥❣❡ t❤❡✐r ❝♦❤❡r❡♥❝❡ ✇❤❡♥ ❡♠♣❧♦②❡❞
t♦❣❡t❤❡r✳
❚♦ ❞❡✈❡❧♦♣ t❤✐s ❞✐s❝✉ss✐♦♥✱ ✐t ✐s ❢✉♥❞❛♠❡♥t❛❧ t❤❛t ✇❡ ❝♦♥s✐❞❡r ✇❤❛t ✐s ✉s✉❛❧❧②
t❛❦❡♥ ❛s t❤❡ ✏♠♦st str✐❦✐♥❣✑ r❡❛s♦♥ ❢♦r ❞✐❛❧❡t❤❡✐s♠✿ t❤❡ ▲✐❛r ♣❛r❛❞♦①✳ ■♥ ❢❛❝t✱
♦♥❡ ♦❢ t❤❡ ♠♦st t❡❧❧✐♥❣ r❡❛s♦♥s ❢♦r ♦♥❡ t♦ ❜❡❝♦♠❡ ❛ ❞✐❛❧❡t❤❡✐st ❝♦♠❡s ❢r♦♠ t❤❡
❞❡s✐r❡ t♦ ❣✐✈❡ ❛ ❞❡✜♥✐t✐✈❡ ❛♥s✇❡r t♦ t❤❡ ▲✐❛r✳
❆❝❝♦r❞✐♥❣ t♦ Pr✐❡st✱ s❡♠❛♥t✐❝
♣❛r❛❞♦①❡s✱ s✉❝❤ ❛s t❤❡ ▲✐❛r✱ s❤♦✇ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❞✐❛❧❡t❤❡✐❛s ✭❢♦r ✐♥st❛♥❝❡✱ s❡❡
Pr✐❡st✱ ❬✾✱ ♣✳✾❪✱ ❬✶✵✱ ♣✳✽✸❪✮✳ ❚❤❡ ❜❛s✐❝ ✐❞❡❛ ✐s t❤❛t t❤❡ ▲✐❛r s❤♦✉❧❞ ❝❛✉s❡ ✉s ♥♦
s♣❡❝✐❛❧ tr♦✉❜❧❡❀ ✐t ✐s ❛s ❣♦♦❞ ❛ ❞❡r✐✈❛t✐♦♥ ❛s ❛♥② ♦t❤❡r✱ ❜❡✐♥❣ ✈❛❧✐❞ ❛♥❞ s♦✉♥❞✳
❚❤❡ ♣r❡♠✐s❡s ❛r❡ ❛❧❧ s❡❡♠✐♥❣❧② tr✉❡ ❛♥❞ ❢♦r t❤❡ ❞✐❛❧❡t❤❡✐st✱ t❤❡ ❝♦♥❝❧✉s✐♦♥
✭❛ ❝♦♥tr❛❞✐❝t✐♦♥✮ ✐s ❥✉st ❛s tr✉❡ ❛s ❛ ❞✐❛❧❡t❤❡✐❛ ♠✉st ❜❡✳
❉✐❛❧❡t❤❡✐s♠✱ t❤❡♥✱
❛❝❝♦✉♥ts ❢♦r ❛ ♣❤❡♥♦♠❡♥♦♥ ♦❢ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡ ✭t❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▲✐❛r✮
✇✐t❤♦✉t t❤❡ ♥❡❡❞ ♦❢ ❛rt✐✜❝✐❛❧ r❡str✐❝t✐♦♥s ♦♥ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡✱ s✉❝❤ ❛s ❚❛rs❦✐✬s
✷
❤✐❡r❛r❝❤✐❡s ♦❢ ❧❛♥❣✉❛❣❡s✳
❇✉t ❤♦✇ ❣♦♦❞ ✐s t❤❛t ❛r❣✉♠❡♥t ❧❡❛❞✐♥❣ ❢r♦♠ t❤❡ ▲✐❛r t♦ ❞✐❛❧❡t❤❡✐s♠❄ ◆♦t✐❝❡
t❤❛t✱ ❛s ✇❡ ❤❛✈❡ ❜r✐❡✢② ❝❤❛r❛❝t❡r✐③❡❞ t❤❡♠✱ t❤❡ ❝♦♥❝❡♣ts ♦❢ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥
❛♥❞ ❣❧✉ts ❛r❡ ❝❧❡❛r❧② ✐♥t❡rt✇✐♥❡❞✳ ■♥ ❢❛❝t✱ ❜♦t❤ ❝♦♥❝❡♣ts ❛r❡ t❛❦❡♥ ❛❧♠♦st ❛s
s②♥♦♥②♠♦✉s ✐♥ ❝✉rr❡♥t ❞✐s❝✉ss✐♦♥s✳ ❚❤❡ ▲✐❛r ✐s ❡✈✐❞❡♥❝❡ ❢♦r ❜♦t❤ ❞✐❛❧❡t❤❡✐❛s
❛♥❞ ❢♦r ❣❧✉ts✱ t❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ❞✐st✐♥❣✉✐s❤ t❤❡♠✱ ✐t s❡❡♠s✳
❆❝❝♦r❞✐♥❣ t♦
Pr✐❡st ❬✾✱ ❈❤❛♣✳✹❪✱ ❞✐❛❧❡t❤❡✐s♠ ✐s ♥♦t ❛ ❝♦♥s✐st❡♥t ❛❝❝♦✉♥t ♦❢ t❤❡ ♣❛r❛❞♦①❡s✱
❞✉❡ t♦ ✐ts ❡♠❜r❛❝❡ ♦❢ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s✱ ❜✉t ✐t ✐s ❛ ❝♦❤❡r❡♥t ✈✐❡✇✱ ✇❤❡r❡ t❤❡
❝♦♥❝❡♣ts ♦❢ ❣❧✉ts ❛♥❞ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛r❡ ♣✉t t♦ ✇♦r❦ ✐♥ ♦r❞❡r t♦ ❢r❡❡ ✉s
✸
❢r♦♠ t❤❡ ❢❡❛r ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥s✳
■t ✐s ♦✉r ♠❛✐♥ ❣♦❛❧ ✐♥ t❤✐s ♣❛♣❡r t♦ ❞✐s❡♥t❛♥❣❧❡ t❤♦s❡ ❝♦♥❝❡♣ts ❛♥❞ ❝❛❧❧ ✐♥t♦
q✉❡st✐♦♥ t❤❡ ❝♦❤❡r❡♥❝❡ ♦❢ ❞✐❛❧❡t❤❡✐s♠✳ ❲❡ s❤❛❧❧ ❛r❣✉❡ t❤❛t t❤❡ ❝♦r❡ ❝♦♥❝❡♣ts
❛r❡ r❡❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✱ ❛♥❞ t❤❛t t❤❡ ▲✐❛r ❝❛♥♥♦t ❜❡ t❛❦❡♥ ❛s ❡✈✐❞❡♥❝❡ ❢♦r t❤❡✐r
❝♦♥♥❡❝t✐♦♥ ❛s ❡st❛❜❧✐s❤❡❞ ❛❜♦✈❡✳ ❋✉rt❤❡r♠♦r❡✱ ✐♥❝♦❤❡r❡♥❝❡ t❤r❡❛t❡♥s ❞✐❛❧❡t❤❡✲
✐s♠ ♦♥❝❡ t❤♦s❡ ❝♦♥❝❡♣ts ❛r❡ ❡①❛♠✐♥❡❞ ♠♦r❡ ❝❧♦s❡❧②✳ ❚❤❡ ♠❛✐♥ r❡s✉❧t ♦❢ s✉❝❤ ❛
✷
❆s Pr✐❡st ❤❛s ♣✉t ✐t✱ ✇❤❛t ✐s ❛t ✐ss✉❡ ✇✐t❤ t❤❡ ▲✐❛r ❛♥❞ r❡❧❛t❡❞ ♣❛r❛❞♦①❡s ✐s t❤❡ ❝♦♥s✐st❡♥❝②
♦❢ t❤❡ ❢❛♠✐❧✐❛r ✐♥❢♦r♠❛❧ ❝♦♥❝❡♣ts ✭❧✐❦❡ t❤❡ ❝♦♥❝❡♣t ♦❢ tr✉t❤ ✐♥ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡✮❀ t❤❡ ❢❡❛s✐❜✐❧✐t②
♦❢ ❝♦♥s✐st❡♥t ❢♦r♠❛❧ t❤❡♦r✐❡s ✐s ♥♦t ❛t ✐ss✉❡ ✭s❡❡ Pr✐❡st ❬✾✱ ❈❤❛♣✳✶❪✮✳
✸
❋♦r ❢✉rt❤❡r r❡❢❡r❡♥❝❡s ♦♥ t❤❡ ❡①♣❡❝t❡❞ ❝♦❤❡r❡♥❝❡ ❝♦♥❝❡r♥✐♥❣ ❞✐❛❧❡t❤❡✐s♠✱ s❡❡ ❛❧s♦ Pr✐❡st
❬✺✱ ♣✳✷✸✾❪✱ ❬✶✷✱ ♣✳✺✵✶✱♣✳✺✵✺❪✱ ❬✾✱ ♣✳✵✻❪✳
✾✵
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
❞✐s❝✉ss✐♦♥ ♠❛② ❜❡ s❡❡♥ ❛s ✇❡❛❦❡♥✐♥❣ t❤❡ r❡❛s♦♥s ❢♦r ♦♥❡ t♦ ❛❞♦♣t ❞✐❛❧❡t❤❡✐s♠✿
✐t ✐s ♥♦t r❡❛❧❧② ❛ ♠♦st ♥❛t✉r❛❧ tr❡❛t♠❡♥t ♦❢ s❡♠❛♥t✐❝ ♣❛r❛❞♦①❡s✱ ✐♥ ♣❛rt✐❝✉❧❛r✱
♥♦t ♦❢ t❤❡ ▲✐❛r ♣❛r❛❞♦①✳ ❚❤❡ ❞✐❛❧❡t❤❡✐st t❤✐♥❦s s❤❡ ✐s ❛❜❧❡ t♦ ❛❝❝♦✉♥t ❢♦r ❢❡❛✲
t✉r❡s ♦❢ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡s t♦ ✇❤✐❝❤ ♦t❤❡r ❛♣♣r♦❛❝❤❡s ❢❛✐❧ t♦ ♠❛❦❡ ❢✉❧❧ ❥✉st✐❝❡✳
❲❤❡♥ t❤❡ ❦♥♦t ❜❡t✇❡❡♥ ❞✐❛❧❡t❤❡✐❛s✱ ❝♦♥tr❛❞✐❝t✐♦♥s✱ ❛♥❞ ❣❧✉ts ✐s ✉♥t✐❡❞✱ t❤✐♥❣s
❛r❡ ♥♦t s♦ ❝❧❡❛r❀ t❤❡ ❞❛t❛ ❢✉r♥✐s❤❡❞ ❜② ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡s ♠❛② ♥♦t ♣r♦✈✐❞❡ ❛♥②
❡✈✐❞❡♥❝❡ ✐♥ ❢❛✈♦r ♦❢ t❤❡ ♣❛rt✐❝✉❧❛r ✈❡rs✐♦♥ ♦❢ ❞✐❛❧❡t❤❡✐s♠ ♠♦st ❝♦♠♠♦♥ ✐♥ t❤❡
❧✐t❡r❛t✉r❡✳ ❆s ✇❡ s❤❛❧❧ ❛r❣✉❡✱ ✐t ♠❛② ❡✈❡♥ ♥♦t ❜❡ s♦ ❡❛s② t♦ ♠❛❦❡ ✐t ❝❧❡❛r ✇❤❛t
tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛♠♦✉♥t t♦✳
❚❤✐s ♣❛♣❡r ✐s str✉❝t✉r❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥ ✇❡ ♣r❡s❡♥t t❤❡ ♠❛✐♥
♠♦t✐✈❛t✐♦♥ ❢♦r ❞✐❛❧❡t❤❡✐s♠ ❝♦♠✐♥❣ ❢r♦♠ t❤❡ ▲✐❛r ♣❛r❛❞♦①✳ ❲❡ ❞✐s❝✉ss t❤❡ ✐❞❡❛
t❤❛t t❤❡ ▲✐❛r ♣r❡s❡♥ts ✉s ✇✐t❤ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛♥❞ ❝❤❛❧❧❡♥❣❡ t❤❡ ❛❜✐❧✐t② ♦❢
❞✐❛❧❡t❤❡✐s♠ t♦ ❞❡❛❧ ✇✐t❤ ♦♥❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ▲✐❛r✱ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r✳ ❚❤❡♥✱ ✐♥
s❡❝t✐♦♥ ✸ ✇❡ ❞✐s❝✉ss ❣❧✉ts✳ ❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✐❞❡❛s ❛r♦✉♥❞ ❣❧✉ts✿ t❤❡ ✐♥❢♦r♠❛❧
♦♥❡✱ ♣r❡s❡♥t❡❞ ❛s t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ▲✐❛r✱ ❛♥❞ t❤❡ ❧♦❣✐❝❛❧ ♦♥❡✱ ❛ t❤✐r❞ tr✉t❤
✈❛❧✉❡ ♦♣❡r❛t✐♥❣ ❛s ❛ s✐♥❣❧❡
s✉✐ ❣❡♥❡r✐s
tr✉t❤✲✈❛❧✉❡✳ ❚❤♦s❡ t✇♦ ✐❞❡❛s ♣♦✐♥t t♦
❞✐✛❡r❡♥t ✐♥t✉✐t✐♦♥s ❛❜♦✉t ❤♦✇ ❣❧✉ts ❜❡❤❛✈❡✳
❆s ✇❡ s❤❛❧❧ s❡❡✱ ❜♦t❤ ❛❝❝♦✉♥ts
♦❢ ❣❧✉ts ❝❛✉s❡ tr♦✉❜❧❡ ❢♦r t❤❡ ❞❡s✐❞❡r❛t❛ ♦❢ ❛♥② ❞✐❛❧❡t❤❡✐st ❛♥s✇❡r t♦ t❤❡ ▲✐❛r✳
❚❤♦s❡ t✇♦ s❡❝t✐♦♥s ♣✉t s♦♠❡ ♣r❡ss✉r❡ ♦♥ t❤❡ ❝♦❤❡r❡♥❝❡ ♦❢ ❞✐❛❧❡t❤❡✐s♠✳
❲❡
❝♦♥t✐♥✉❡ ✐♥ s❡❝t✐♦♥ ✹ ✇✐t❤ ❛♥ ❛r❣✉♠❡♥t t♦ t❤❡ ❡✛❡❝t t❤❛t ❡✈❡♥ t❤❡ ❞✐❛❧❡t❤❡✐st
♠✉st ❛❞♠✐t t❤❛t t❤❡ ♣❛r❛❞♦① ♦❢ t❤❡ ▲✐❛r ❡♥❞s ✉♣ ✇✐t❤ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ✇❤✐❝❤ ✐s
✐♥❞❡❡❞ ❛ ❢❛❧s❡ ❝♦♥❝❧✉s✐♦♥✳ ❙♦✱ t❤❡ ❞✐❛❧❡t❤❡✐st tr❡❛t♠❡♥t ❞♦❡s ♥♦t ✇♦r❦ ❛♥❞ t❤❡
♣❛r❛❞♦① ✐s st✐❧❧ ❛ ♣r♦❜❧❡♠✱ ❡✈❡♥ ❢♦r t❤❡ ❞✐❛❧❡t❤❡✐st✳ ❲❡ ❝♦♥❝❧✉❞❡ ✐♥ s❡❝t✐♦♥ ✺✳
✷ ❲❤❡r❡ ❞♦❡s t❤❡ ▲✐❛r ❧❡❛❞ ✉s t♦❄
❲❤❛t ❞♦❡s ❞✐❛❧❡t❤❡✐s♠✱ ❛s ♣r❡✈✐♦✉s❧② ❝❤❛r❛❝t❡r✐③❡❞✱ ✐♠♣❧② ❛♥❞ ❤♦✇ ✐s ✐t ♠♦t✐✲
✈❛t❡❞❄ ❚♦ ❛♥s✇❡r t❤♦s❡ q✉❡st✐♦♥s✱ ✇❡ ♠❛② ❛tt❛✐♥ ♦✉rs❡❧✈❡s t♦ t❤❡ ♠❛✐♥ ♠♦t✐✲
✈❛t✐♦♥ ❢♦r ❞✐❛❧❡t❤❡✐s♠✱ t❤❡ ▲✐❛r ♣❛r❛❞♦①✱ ❛♥❞ Pr✐❡st✬s ❛♥s✇❡r t♦ ✐t✳ ■t ❝♦✉❧❞ ❜❡
s❛✐❞ t❤❛t t❤❡ ❛♥s✇❡r ❛♠♦✉♥ts t♦ st❛t✐♥❣ t❤❡ ▲✐❛r ❛♥❞ t❤❡♥ s✐♠♣❧② t❛❦✐♥❣ t❤❡
❝♦♥❝❧✉s✐♦♥ ❛s ✐t ✐s✿ ❢r♦♠ ❛ s♦✉♥❞ ❛r❣✉♠❡♥t ✇❡ r❡❛❝❤ t❤❡ ❢❛❝t t❤❛t ②❡s✱ t❤❡r❡
❛r❡ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ✭s❡❡ ❛❧s♦ Pr✐❡st ✭❬✶✷❪✱ ❬✾❪✱ ❬✶✵❪✱ ❬✶✶❪✮✳ ❲❡ ❛r❡ ❣♦✐♥❣ t♦
❝♦♥s✐❞❡r t✇♦ ✈❡rs✐♦♥s ♦❢ t❤✐s ♣❛r❛❞♦①✿ t❤❡ ✏s✐♠♣❧❡✑ ✭♦r ✏♦r❞✐♥❛r②✑✮ ▲✐❛r ❛♥❞ t❤❡
✏❡①t❡♥❞❡❞✑ ✭♦r ✏str❡♥❣t❤❡♥❡❞✑✮ ▲✐❛r❀ ✇❡ s❤❛❧❧ ❛r❣✉❡ t❤❛t ✐t ✐s ♥♦t t❤❛t s✐♠♣❧❡ t♦
❛❝❝❡♣t t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤♦s❡ ❛r❣✉♠❡♥ts ❛s ❞✐❛❧❡t❤❡✐❛s✳ ❲❤❛t t❤♦s❡ ♣❛r❛❞♦①❡s
t❡❧❧ ✉s ♠❛② ❝❛✉s❡ tr♦✉❜❧❡ ❢♦r t❤❡ ❞✐❛❧❡t❤❡✐sts✬ ❛❝❝♦✉♥t ♦❢ s❡♠❛♥t✐❝ ♣❛r❛❞♦①❡s✳
■♥t✉✐t✐✈❡❧②✱ ❛❧❧ t❤❛t ✐s r❡q✉✐r❡❞ t♦ ❞❡r✐✈❡ t❤❡ ▲✐❛r ❛r❡ s♦♠❡ s✐♠♣❧❡ r❡s♦✉r❝❡s✿
❜❛s✐❝ ❧♦❣✐❝ ❛♥❞ ❛ tr✉t❤ ♣r❡❞✐❝❛t❡
T
✐♥ t❤❡ ♦❜❥❡❝t ❧❛♥❣✉❛❣❡ s❛t✐s❢②✐♥❣ t❤❡
s❝❤❡♠❛ ❢♦r ❡✈❡r② s❡♥t❡♥❝❡ ♦❢ t❤❡ ❧❛♥❣✉❛❣❡✳
▲❡t ✉s st❛rt ✇✐t❤ t❤❡
✐ts❡❧❢ t❤❛t ✐t ✐s ❢❛❧s❡ ✿
s✐♠♣❧❡ ▲✐❛r✱
❛ s❡♥t❡♥❝❡
L
t❤❛t✱ ✐♥t✉✐t✐✈❡❧②✱
T✲
s❛②s ♦❢
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✾✶
L ↔ F ✭_L^✮
❍❡r❡ t❤❡ ❛♥❣❧❡ ❜r❛❝❦❡ts ❛r❡ ❛ ♥❛♠❡✲❢♦r♠✐♥❣ ❞❡✈✐❝❡ ❛♥❞ ❵F ✬ ✐s t❤❡ ❢❛❧s✐t②
♣r❡❞✐❝❛t❡ ✉♥❞❡rst♦♦❞ ✐♥ t❤❡ ✉s✉❛❧ ✇❛②✿ ❵❢❛❧s❡✬ ✭✐✳❡✳ F ✭_L^✮✮ ♠❡❛♥s ❵❤❛s ❛ tr✉❡
♥❡❣❛t✐♦♥✬ ✭✐✳❡✳ T ✭_¬L^✮✳ ❆ss✉♠✐♥❣ t❤❡ ✐♥st❛♥❝❡ ♦❢ t❤❡ T ✲s❝❤❡♠❛ ❛♣♣❧✐❡❞ t♦ L✱
✭T ✭_L^✮ ↔ L✮✱ ✇✐t❤ ❵T ✬ ❜❡✐♥❣ t❤❡ tr✉t❤ ♣r❡❞✐❝❛t❡✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ T ✭_L^✮
↔ F ✭_L^✮✳ ❋r♦♠ t❤✐s✱ ✉❧t✐♠❛t❡❧②✱ ✇❡ ❤❛✈❡ ❛s ❛ ❝♦♥❝❧✉s✐♦♥ t❤❛t T ✭_L^✮ ∧ F ✭_L^✮✳
❚❤❛t ✐s✱ t❤❡ ▲✐❛r s❡♥t❡♥❝❡ L ✐s ❜♦t❤ tr✉❡ ❛♥❞ ❢❛❧s❡✳
❚❤❡ r❡❛s♦♥✐♥❣ ♣r♦❝❡ss ✭t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❢❛❧s❡❤♦♦❞✱ t❤❡ T ✲s❝❤❡♠❛✱ ❡t❝✳✮ ✐s
❞✐❛❧❡t❤❡✐❝❛❧❧② ❛❝❝❡♣t❛❜❧❡ ❛♥❞✱ ♦❢ ❝♦✉rs❡✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ✭t❤❡ ❞✐❛❧❡t❤❡✐❛✮✱ ✐s ❛❝✲
❝❡♣t❛❜❧❡ ❢♦r ❞✐❛❧❡t❤❡✐sts ❛s ✇❡❧❧✳ ❙♦✱ ❛❝❝♦r❞✐♥❣ t♦ ❞✐❛❧❡t❤❡✐sts✱ t❤✐s ❛r❣✉♠❡♥t
s❤♦✇s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❞✐❛❧❡t❤❡✐❛s✳
❍♦✇❡✈❡r✱ t❤❛t ✐s ♥♦t t❤❡ ♦♥❧② ❢♦r♠ ♦❢ t❤❡ ▲✐❛r t❤❛t Pr✐❡st ❝❛❧❧s ❢♦rt❤ ✐♥ ♦r❞❡r
t♦ s✉♣♣♦rt ❤✐s ✈✐❡✇✳ ❆❝❝♦r❞✐♥❣ t♦ Pr✐❡st ❛♥❞ ♦t❤❡r ❞✐❛❧❡t❤❡✐sts✱ ❡✈❡r② ❝♦♥s✐s✲
t❡♥t ❛tt❡♠♣t t♦ tr❡❛t t❤❡ s❡♠❛♥t✐❝ ♣❛r❛❞♦①❡s ❧❡❛❞s t♦ ❡①t❡♥❞❡❞ ▲✐❛r ♣❛r❛❞♦①❡s
✭s❡❡✱ ❢♦r ✐♥st❛♥❝❡✱ ❬✻✱ ♣✳✶✺✽❪✱ ❬✾✱ ❝❤❛♣✳✶❪❀ ❬✶✷✱ ♣♣✳✺✵✽✕✺✶✵❪❀ ❬✶✶✱ s❡❝✳✺✳✶❪ ❛♥❞ t❤❡
r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✮✳ ❋♦r ✐♥st❛♥❝❡✱ ✐♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❣❛♣ ❛♣♣r♦❛❝❤ ✖ t❤❡ ✈✐❡✇
♣r♦♣♦s✐♥❣ t❤❛t t❤❡ ▲✐❛r s❡♥t❡♥❝❡ L ✐s ♥❡✐t❤❡r tr✉❡ ♥♦r ❢❛❧s❡ ✖✱ Pr✐❡st ❝❛❧❧s ✉♣♦♥
t❤❡ ❡①t❡♥❞ ▲✐❛r✿ ❛ s❡♥t❡♥❝❡ L t❤❛t✱ ✐♥t✉✐t✐✈❡❧②✱ s❛②s ♦❢ ✐ts❡❧❢ t❤❛t ✐t ✐s ♥♦t tr✉❡ ✿
L ↔ ¬T ✭_L^✮
❇② r❡❛s♦♥✐♥❣ s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ✐♥ t❤❡ s✐♠♣❧❡ ▲✐❛r✱ ✐♥ t❤✐s ❝❛s❡✱ ✇❡ ❤❛✈❡ ❛s
❝♦♥❝❧✉s✐♦♥ t❤❛t T ✭_L^✮ ∧¬T ✭_L^✮✳ ❚❤❛t ✐s✱ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r s❡♥t❡♥❝❡ L ✐s ❜♦t❤
tr✉❡ ❛♥❞ ♥♦t tr✉❡✳
❇✉t ✇❤② ❛r❡ ✇❡ ✐♥s✐st✐♥❣ ✐♥ t❤❡s❡ t✇♦ ✈❡rs✐♦♥s ♦❢ t❤❡ ▲✐❛r❄ ❚❤❡ ❝♦♥❝❧✉s✐♦♥
✐♥ ❡❛❝❤ ❝❛s❡ ✐s♥✬t ❜❛❞ ❡♥♦✉❣❤ ❢♦r t❤❡ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✐❛♥❄ ❆♥❞ ✐s♥✬t ✐t ❛❧❧ t❤❡ s❛♠❡
❢♦r t❤❡ ❞✐❛❧❡t❤❡✐st❄ ■♥ ❢❛❝t✱ ❢♦r t❤❡ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✐❛♥ t❤❡ ❝♦♥❝❧✉s✐♦♥s ♦❢ ❜♦t❤
❛r❣✉♠❡♥ts ❛r❡ ❡q✉❛❧❧② ❜❛❞✱ t❤❡② ❛r❡ ❡q✉❛❧❧② ❢❛❧s❡ ✭✇❡ s❤❛❧❧ r❡t✉r♥ t♦ t❤✐s ♣♦✐♥t
❧❛t❡r✮✱ s♦ s❤❡ ♠✉st ❛✈♦✐❞ ❛❝❝❡♣t✐♥❣ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❛r❣✉♠❡♥t✳ ❍♦✇❡✈❡r✱
❢♦r t❤❡ ❞✐❛❧❡t❤❡✐st ❜♦t❤ ❛r❣✉♠❡♥ts ❛r❡ ♥♦t ❡q✉❛❧❧② ❣♦♦❞❀ ♦r✱ ❛t ❧❡❛st✱ ✐t ✐s ♥♦t
s♦ ♦❜✈✐♦✉s t❤❛t t❤❡② ❡①❤✐❜✐t t❤❡ s❛♠❡ ❦✐♥❞ ♦❢ ♣r♦❜❧❡♠✳ ❆❝❝♦r❞✐♥❣ t♦ Pr✐❡st✬s
♦✇♥ st❛♥❞❛r❞s✱ t❤❡ ♣❛r❛❞♦①❡s ❛r❡ ❞✐✛❡r❡♥t✳ ▲❡t ✉s ❝❤❡❝❦✳
❇♦t❤ ❛r❣✉♠❡♥ts ✇♦✉❧❞ s❤♦✇ t❤❡ s❛♠❡ t❤✐♥❣ ✐❢ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥
T ✭_¬L^✮ ❛♥❞ ¬T ✭_L^✮ ✇❡r❡ ❡st❛❜❧✐s❤❡❞✱ ❛s ✐t ✐s t❤❡ ❝❛s❡ ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ ❚❤❡♥✱
❡✈❡♥ t❤❡✐r ❞❡✜♥✐t✐♦♥s ✇♦✉❧❞ ❜❡ ❡q✉✐✈❛❧❡♥t✳ Pr✐❡st ✭❬✾✱ ♣✳✼✵❪✮ s❡♣❛r❛t❡s t❤❡ ❡q✉✐✈✲
❛❧❡♥❝❡ ✐♥t♦ t✇♦ ✐♠♣❧✐❝❛t✐♦♥s✿
❊①❤❛✉st✐♦♥✿
❊①❝❧✉s✐♦♥✿
¬T ✭_L^✮ → T ✭_¬L^✮
T ✭_¬L^✮ → ¬T ✭_L^✮
✾✷
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
❚♦ ❜❡❣✐♥ ✇✐t❤ t❤❡ ✜rst ♦♥❡✱ ❡①❤❛✉st✐♦♥ s❛②s t❤❛t ✐❢ ❛ s❡♥t❡♥❝❡ ✐s ♥♦t tr✉❡✱
t❤❡♥ ✐t ✐s ❢❛❧s❡✳ ❙♦✱ ❚r✉t❤ ❛♥❞ ❋❛❧s✐t② ❡①❤❛✉st t❤❡ ♣♦ss✐❜✐❧✐t✐❡s ♦❢ tr✉t❤ ✈❛❧✉❡s✱
❛♥❞ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡ t❤❡r❡ ❛r❡ ♥♦ tr✉t❤ ✈❛❧✉❡ ❣❛♣s ✭✐✳❡✳ s❡♥t❡♥❝❡s ♥❡✐t❤❡r tr✉❡
♥♦r ❢❛❧s❡✮✳ ❊①❝❧✉s✐♦♥✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ s❛②s t❤❛t ✇❤❡♥❡✈❡r ❛ s❡♥t❡♥❝❡ ✐s ❢❛❧s❡✱
t❤❡♥ ✐t ✐s ♥♦t ❛❧s♦ tr✉❡✳ ■t ❝❧❡❛r❧② ❢♦r❜✐❞s ❛ s❡♥t❡♥❝❡ ❢r♦♠ ❜❡✐♥❣ ❜♦t❤ tr✉❡ ❛♥❞
❢❛❧s❡✱ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ ♦❢ ❜❡❛r✐♥❣ ❛ ❣❧✉t ❛s ✐ts tr✉t❤ ✈❛❧✉❡✳ ❚❤❡ ❣❛♣ t❤❡♦r✐st
r❡❥❡❝ts ❡①❤❛✉st✐♦♥ ✭♥♦t ❜❡✐♥❣ tr✉❡✱ ❛ s❡♥t❡♥❝❡ ♠❛② ❛❧s♦ ♥♦t ❜❡ ❢❛❧s❡✮✱ ✇❤✐❧❡ t❤❡
❣❧✉t t❤❡♦r✐sts✱ ❧✐❦❡ Pr✐❡st✱ t②♣✐❝❛❧❧② ❛❝❝❡♣t ❡①❤❛✉st✐♦♥ ✇❤✐❧❡ r❡❥❡❝t✐♥❣ ❡①❝❧✉s✐♦♥✳
❙♦✱ ✇✐t❤ t❤❛t ❞✐st✐♥❝t✐♦♥ ✐♥ ❤❛♥❞✱ ❛♥❞ t❛❦✐♥❣ Pr✐❡st✬s ❛r❣✉♠❡♥ts ❛❣❛✐♥st
❡①❝❧✉s✐♦♥ ✭s❡❡ Pr✐❡st ✭❬✾✱ ❝❤❛♣✳✹❪✮✱ ✇❡ ❝❧❡❛r❧② s❡❡ ✐t t❤❛t t❤❡r❡ ❛r❡ t✇♦ ❦✐♥❞s ♦❢
▲✐❛r ❛r❣✉♠❡♥ts ✭✇❡ s❤❛❧❧ r❡t✉r♥ t♦ Pr✐❡st✬s ❛r❣✉♠❡♥ts ❛❣❛✐♥st ❡①❝❧✉s✐♦♥ s♦♦♥✮✳
Pr✐❡st ❡✈❡♥ ❣♦❡s ♦♥ t♦ ❝❧❛✐♠ t❤❛t t❤♦s❡ ❝♦♥❝❧✉s✐♦♥s ♦❢ t❤❡ s✐♠♣❧❡ ▲✐❛r ❛♥❞ ♦❢
t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ❤❛✈❡ ❞✐st✐♥❝t ♣r♦♣❡rt✐❡s✳ ■♥ ❢❛❝t✱ ❛s ❛ ✜rst st❡♣✱ ✐t ✐s ♣♦ss✐❜❧❡
t♦ s❤♦✇ t❤❛t ❢r♦♠ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r
❡①✐st❡♥t✐❛❧ q✉❛♥t✐✜❡r
∃x(T (x) ∧ ¬T (x))
T ✭_L^✮ ∧ ¬T ✭_L^✮✱
t❤❡
❢♦❧❧♦✇s✳ ❆❧s♦✱ ❢r♦♠ t❤❡ ❧❛✇ ♦❢ ❡①❝❧✉❞❡❞
∀x(T (x)∨
¬T (x))✱ ❛♥❞ ✇✐t❤ t❤❡ ❞❡ ▼♦r❣❛♥ ❧❛✇ ✭✐❞❡♠✮✱ ✇❡ ❞❡r✐✈❡ ¬∃x(T (x) ∧ ¬T (x))✳
❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢r♦♠ t❤❡ s✐♠♣❧❡ ▲✐❛r ✇❡ ❝❛♥ ❞❡r✐✈❡ ∃x(T (x) ∧ F (x))✱ ❜✉t
✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ♥❡❣❛t✐♦♥ ¬∃x(T (x) ∧ F (x)) ❝❛♥♥♦t ❜❡ ❞❡r✐✈❡❞ ✇✐t❤♦✉t t❤❡
♠✐❞❞❧❡ ✭✇❤✐❝❤ ✐s ❞✐❛❧❡t❤❡✐❝❛❧❧② ❛❝❝❡♣t❛❜❧❡✱ s❡❡ Pr✐❡st ❬✾❪✱ ❬✶✵❪✮ ✇❡ ❤❛✈❡
❡①❝❧✉s✐♦♥ ♣r✐♥❝✐♣❧❡ ✭s❡❡ Pr✐❡st ✭❬✾✱ ♣✳✼✷❪✮✳
■♥ t❤✐s s❡♥s❡✱ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✐s
✏♠♦r❡ ❝♦♥tr❛❞✐❝t♦r②✑ t❤❛♥ t❤❡ s✐♠♣❧❡ ▲✐❛r ✭t❤❛t ✐s Pr✐❡st✬s ♦✇♥ ❝♦♥❝❧✉s✐♦♥✮✳
❍♦✇❡✈❡r✱ ✐❢ t❤❛t ❞✐✛❡r❡♥❝❡ ♠❛❦❡s s❡♥s❡ ❛♥❞ ❜♦t❤ ❛r❣✉♠❡♥ts ❛r❡ t❛❦❡♥ t♦ ❜❡
s♦✉♥❞ ✐♥ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡✱ t❤❡♥ t❤❡r❡ ✐s ❝❧❡❛r❧② ❛ ❣❛♣ ✐♥ t❤❡ ❞✐❛❧❡t❤❡✐sts✬ st♦r②
❛❜♦✉t t❤❡ ▲✐❛r✳ ❚♦ ❜❡❣✐♥ ✇✐t❤ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r✱ ✈✐③✳
∧ ¬T ✭_L^✮✱
✐t ✐s ❝❧❡❛r t❤❛t s✉❝❤ ❛ s❡♥t❡♥❝❡ ✐s ♥♦t ✐ts❡❧❢ s❛②✐♥❣ t❤❛t
L
T ✭_L^✮
✐s ❛ ❣❧✉t✳
■♥❞❡❡❞✱ ✐t ✐s ❛ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❛ s❡♥t❡♥❝❡ tr✉❡ ❛♥❞ ♥♦t tr✉❡✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t
L
✐s ❛ ❣❧✉t ✭❞✉❡ t♦ ❡①❤❛✉st✐♦♥✮✱ ❜✉t ❣✐✈❡♥ t❤❡ ❛❜♦✈❡ ❝♦♥s✐❞❡r❛t✐♦♥s ❜❛♥♥✐♥❣
❡①❝❧✉s✐♦♥✱ t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❛ s❡♥t❡♥❝❡ tr✉❡ ❛♥❞ ❢❛❧s❡ ✭❛ ❣❧✉t ❝❛s❡✮ ❞♦❡s ♥♦t
✐♠♣❧② ❛ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❛ s❡♥t❡♥❝❡ tr✉❡ ❛♥❞ ♥♦t tr✉❡✳ ❚❤✉s✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢
✹
t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✐s ♥♦t ❛ ❣❧✉t✳
t❤❡ s❡♥t❡♥❝❡
T ✭_L^✮ ∧ ¬T ✭_L^✮
▲❡t ✉s ♣✉t ✐t ❛♥♦t❤❡r ✇❛②✿ ❛❢t❡r ❞❡r✐✈✐♥❣
❛s t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r✱ ✇❤❛t ✇❡
❤❛✈❡ ❛s t❤❡ ❧❛st st❡♣ ✐♥ t❤❡ ❞❡r✐✈❛t✐♦♥ ✐s ♥♦t t❤❛t
L
✐s tr✉❡ ❛♥❞ ❢❛❧s❡❀ r❛t❤❡r✱
L
✐s tr✉❡ ❛♥❞ ♥♦t tr✉❡✳ ■t ❝❧❡❛r❧② ❤❛s t❤❡ ❢♦r♠ r❡q✉✐r❡❞ t♦ ❜❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉t
♥♦t t❤❡ ❢♦r♠ ♦❢ ❛ ❣❧✉t ✭tr✉❡ ❛♥❞ ❢❛❧s❡✮✳ ❚❤❛t ✐s✱ t❤❡ s✐t✉❛t✐♦♥ ✐♥ t❤❡ ❡①t❡♥❞❡❞
▲✐❛r ♠❛② ❜❡ ❞❡♣✐❝t❡❞ ❛s ❢♦❧❧♦✇s✿
✹
1. L ↔ ¬T ✭_L^✮
❊①t❡♥❞❡❞ ▲✐❛r s❡♥t❡♥❝❡
✳
✳
✳
✳
✳
✳ ✭❙t❛♥❞❛r❞ ❞❡r✐✈❛t✐♦♥✮
n. T ✭_L^✮ ∧ ¬T ✭_L^✮
❈♦♥❝❧✉s✐♦♥
❇❡s✐❞❡s✱ ❛s ✇❡ s❤❛❧❧ ❛r❣✉❡ ✐♥ s❡❝t✐♦♥ ✸✱ t♦ ❝♦♥s✐❞❡r t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ❛s ❛ ❣❧✉t ✇✐❧❧ ❤❛✈❡
s♦♠❡ ❝♦♥s❡q✉❡♥❝❡s ❢♦r t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❛ ❣❧✉t t❤❛t ❛r❡ ♥♦t s♦ ❣♦♦❞ ❢♦r ❞✐❛❧❡t❤❡✐s♠✳
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✾✸
■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❛t L ✐s tr✉❡ ❛♥❞ ❢❛❧s❡ ♠♦r❡ st❡♣s ❛r❡ r❡q✉✐r❡❞ ✐♥ t❤❡
❞❡r✐✈❛t✐♦♥✱ ✉s✐♥❣ ❡①❤❛✉st✐♦♥✳ ❙♦✱ t❤❡ ❞❡r✐✈❛t✐♦♥ ✇♦✉❧❞ ❜❡ s♦♠❡ ❧✐♥❡s ❧♦♥❣❡r✿
1. L ↔ ¬T ✭_L^✮
❊①t❡♥❞❡❞ ▲✐❛r s❡♥t❡♥❝❡
✳✳
✳✳
✳
✳ ✭❙t❛♥❞❛r❞ ❞❡r✐✈❛t✐♦♥✮
n. T ✭_L^✮ ∧ ¬T ✭_L^✮
❖r✐❣✐♥❛❧ ❝♦♥❝❧✉s✐♦♥
n + 1. T ✭_L^✮
❋r♦♠ n
n + 2. ¬T ✭_L^✮
❋r♦♠ n
n + 3. T ✭_¬L^✮
❊①❤❛✉st✐♦♥ ✐♥ n + 2
n + 4. T ✭_L^✮∧T ✭_¬L^✮ ❋r♦♠ n + 1 ❛♥❞ n + 3
❍♦✇❡✈❡r✱ ❛❞❞✐♥❣ t❤♦s❡ ❢✉rt❤❡r st❡♣s ✐♥ t❤❡ st❛♥❞❛r❞ ❞❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❡①✲
t❡♥❞❡❞ ▲✐❛r ✇♦♥✬t ❤❡❧♣✿ t❤❡ ♦r✐❣✐♥❛❧ tr✉❡ ❛♥❞ ♥♦t✲tr✉❡ s❡♥t❡♥❝❡ r❡♠❛✐♥s t❤❡r❡
✭✐♥ t❤❡ n✲t❤ st❡♣ ✐♥ t❤❡ ❛❜♦✈❡ s❦❡t❝❤ ♦❢ ❛ ❞❡r✐✈❛t✐♦♥✮✳ ❆s ❢❛r ❛s ✇❡ ❦♥♦✇✱ t❤❡
❞✐❛❧❡t❤❡✐sts r❡❥❡❝t✐♥❣ ❡①❝❧✉s✐♦♥ ❤❛✈❡ ♥♦t ②❡t ❡st❛❜❧✐s❤❡❞ t❤❡ tr✉t❤ ✈❛❧✉❡ ♦❢ ❛
❝♦♥❥✉♥❝t✐♦♥ ❜❡t✇❡❡♥ tr✉❡ ❛♥❞ ♥♦t✲tr✉❡ s❡♥t❡♥❝❡s✳ ❙♦✱ t❤❡r❡ ✐s ❛ s❡♥t❡♥❝❡ ✐♥ t❤❡
❞❡r✐✈❛t✐♦♥ ❜❡❛r✐♥❣ ❛ ♠②st❡r✐♦✉s tr✉t❤ ✈❛❧✉❡✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❣❧✉tt② s❡♥t❡♥❝❡s✱ ✇❡
❛t ❧❡❛st t❤✐♥❦ ♦❢ t❤❡ s❡♥t❡♥❝❡ ❛s ❜❡✐♥❣ tr✉❡✳ ■♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡
❡①t❡♥❞❡❞ ▲✐❛r✱ t❤❡r❡ ✐s ♥♦ ❝❧❡❛r tr✉t❤ ✈❛❧✉❡ t♦ ❜❡ ❛ttr✐❜✉t❡❞✳ ❇✉t t❤❡♥✱ ❤♦✇
❝❛♥ ✇❡ ❡✈❡♥ s❛② t❤❛t t❤❡ ❞❡r✐✈❛t✐♦♥ ✐s s♦✉♥❞❄
●✐✈❡♥ t❤❛t✱ ♦✉r ✜rst ♣♦✐♥t ❝♦♥❝❡r♥✐♥❣ t❤❡ ❞✐✛❡r❡♥t ✈❡rs✐♦♥s ♦❢ t❤❡ ♣❛r❛❞♦① ✐s
t❤❛t t❤❡r❡ ❛r❡ ❝♦♥tr❛❞✐❝t✐♦♥s t❤❛t ❛r❡ ♥♦t ❣❧✉ts❀ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ ❡①t❡♥❞❡❞ ▲✐❛r
✐s ❛♥ ❡①❛♠♣❧❡✳ ❚❤✐s ✐s ❡✈✐❞❡♥❝❡ t♦ t❤❡ ❢❛❝t t❤❛t t❤❡ s②♥t❛❝t✐❝❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥
♦❢ ❝♦♥tr❛❞✐❝t✐♦♥ ♠❛② ❜❡ s❛t✐s✜❡❞ ❜② ❛ ❢♦r♠✉❧❛✱ ✇❤✐❧❡ t❤❛t ❢♦r♠✉❧❛ ✐s ♥♦t ❝❧❡❛r❧②
❛ ❣❧✉t✳ ■♥ t❤✐s ❝❛s❡✱ ❝♦♥tr❛❞✐❝t✐♦♥s ❛♥❞ ❣❧✉ts ❞♦ ♥♦t ❛❧✇❛②s ❝♦♠❡ t♦❣❡t❤❡r✳ ❲❡
❦♥❡✇ t❤❛t ❛❧r❡❛❞②✱ ❢♦r ✐❞❡❛❧❧② ♥♦t ❡✈❡r② ❝♦♥tr❛❞✐❝t✐♦♥ ✐s ❛ ❞✐❛❧❡t❤❡✐❛ ✭s♦ ♥♦t
❡✈❡r② ❝♦♥tr❛❞✐❝t✐♦♥ ✐s ❛❧s♦ ❛ ❣❧✉t✮✱ ❜✉t ✐♥ t❤✐s ❝❛s❡ t❤❡r❡ ✐s ♠✉❝❤ ♠♦r❡ ✐♥✈♦❧✈❡❞✿
t❤❡r❡ ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t ♣r✐♠❛ ❢❛❝✐❡ ❝❛♥♥♦t ❜❡ ❞❡❛❧t ✇✐t❤ ❞✐❛❧❡t❤❡✐❝❛❧❧②✱
❣✐✈❡♥ t❤❛t t❤❡ ❛❝❝❡♣t❛❜❧❡ ❝♦♥tr❛❞✐❝t✐♦♥s s❤♦✉❧❞ ❛❧❧ ❜❡ ❣❧✉ts✳ ❖✉r ❛r❣✉♠❡♥t
✐♥ ✇❤❛t ❢♦❧❧♦✇s ✐s t❤❛t ✇❤✐❧❡ ❞✐❛❧❡t❤❡✐sts ❤❛✈❡ ❛ st♦r② t♦ t❡❧❧ ❛❜♦✉t ❤♦✇ t♦
❞❡❛❧ ✇✐t❤ t❤❡ s✐♠♣❧❡ ▲✐❛r✱ t❤❡② ❤❛✈❡ ♥♦t ♣r♦✈✐❞❡❞ s✉❝❤ ❛ st♦r② ❛s t♦ ❤♦✇ t♦
t❛♠❡ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✭❛t ❧❡❛st ♥♦t ✇❤❡♥ ❡①❝❧✉s✐♦♥ ✐s ❛❜s❡♥t✮✳
❋✉rt❤❡r♠♦r❡✱ ❣✐✈❡♥ t❤❛t t❤❡ ❡①t❡♥❞❡❞ ✈❡rs✐♦♥ ✐s ♥♦t ❡q✉✐✈❛❧❡♥t t♦ t❤❡ s✐♠♣❧❡ ♦♥❡
✭❛s Pr✐❡st ❤✐♠s❡❧❢ ❤❛s ❛r❣✉❡❞✮✱ ✇❡ ❝❛♥♥♦t s✐♠♣❧② ❛❞♦♣t t❤❡ tr❡❛t♠❡♥t ❣✐✈❡♥ t♦
t❤❡ s✐♠♣❧❡ ▲✐❛r ❛♥❞ ✉s❡ ✐t t♦ ❞❡❛❧ ✇✐t❤ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r❀ t❤❡ ❝❛s❡s ❛r❡ ❞✐✛❡r❡♥t✳
❙♦✱ t❤❡ ❢❛❝t t❤❛t ♦♥❡ ♦❢ t❤❡ ❝♦♥❝❧✉s✐♦♥s ♦❢ ♦♥❡ ♦❢ t❤❡ ♣❛r❛❞♦①❡s ❡♥t❛✐❧s ❛ s❡♥t❡♥❝❡
t❤❛t ✐s ❥✉st ❧✐❦❡ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ♦t❤❡r s❤♦✉❧❞ ♥♦t ❝♦♥❢✉s❡ ✉s ✐♥t♦ t❤✐♥❦✐♥❣
t❤❛t t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡♠ ✐s t❤❡ s❛♠❡✳ ❚❤❡r❡ s❡❡♠s t♦ ❜❡ t✇♦ ♦♣t✐♦♥s ❢♦r t❤❡
❞✐❛❧❡t❤❡✐st✳ ▲❡t ✉s s❡❡✳
❚❤❡ ✜rst ♦♥❡ ❝♦♥s✐sts ✐♥ ❞❡♥②✐♥❣ t❤❛t t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r
✐s ❛❝❝❡♣t❛❜❧❡✳ ❚❤❛t ✐s✱ ✇❡ s❤♦✉❧❞ ❞✐s❝r✐♠✐♥❛t❡ ❜❡t✇❡❡♥ t❤❡ ✈❡rs✐♦♥s ♦❢ t❤❡
♣❛r❛❞♦① ❛♥❞ ❞❡❡♠ t❤❡ ❡①t❡♥❞❡❞ ✈❡rs✐♦♥ ❛s s♦♠❡❤♦✇ ✉♥❛❝❝❡♣t❛❜❧❡ ♠❛✐♥❧② ❞✉❡
✾✹
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
t♦ t❤❡ tr✉t❤ ✈❛❧✉❡ ✏✉♥tr✉❡✑ t❤❛t ✐♥✜❧tr❛t❡s t❤r♦✉❣❤ ✐t✳ ❖❜✈✐♦✉s❧②✱ t❤❛t ♣♦s❡s
❛♥ ✉♥❞❡s✐r❛❜❧❡ ❛s②♠♠❡tr② ✇✐t❤ t❤❡ tr❡❛t♠❡♥t ♦❢ t❤❡ s✐♠♣❧❡ ▲✐❛r✱ ❧❡❛✈✐♥❣ t❤❡
❞✐❛❧❡t❤❡✐st ✇✐t❤ ♥♦ ❛♥s✇❡r t♦ ❛♥ ✐♠♣♦rt❛♥t s❡♠❛♥t✐❝❛❧ ♣❛r❛❞♦①✱ ❛♥❞ t❤❡ ♠♦✈❡
✐s ❝❧❡❛r❧② ✉♥♠♦t✐✈❛t❡❞ ❢♦r ❛ ❞✐❛❧❡t❤❡✐st✳ ❙♦✱ t❤✐s ✐s ❛ ♥♦♥✲st❛rt❡r✳
❚❤❡ s❡❝♦♥❞ ♦♣t✐♦♥ ❝♦♥s✐sts ✐♥ ❛ss✉♠✐♥❣ t❤❛t t❤✐s ❝♦♥tr❛❞✐❝t✐♦♥ ✐s ✐♥❞❡❡❞
❛❝❝❡♣t❛❜❧❡✳
❚❤✐s ❝❤♦✐❝❡ ✇✐❧❧ ❤❛✈❡ ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ✉t♠♦st ✐♠♣♦rt❛♥❝❡ t❤❛t
✇❡ ❤❛✈❡ ❛❧r❡❛❞② ❤✐♥t❡❞ ❛t✿ t❤❡ t②♣✐❝❛❧ ❛r❣✉♠❡♥t ❣r❛♥t✐♥❣ t❤❛t ❛ ❝♦♥tr❛❞✐❝t✐♦♥
❞♦❡s ♥♦t ❧❡❛❞ t♦ tr✐✈✐❛❧✐t② ✇✐❧❧ ♥♦t ✇♦r❦ ❢♦r t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r
✭t❤❛t ✐s✱ t❤❛t t❤❡ s♦✲❝❛❧❧❡❞ ❡①♣❧♦s✐♦♥ ❧❛✇ ❞♦❡s ♥♦t ♦❜t❛✐♥✮✳ ■♥ ❢❛❝t✱ t❤❡ ✉s✉❛❧
tr✐❝❦ ❢♦r ❜❛♥♥✐♥❣ ❡①♣❧♦s✐♦♥ ❝♦♥s✐sts ✐♥ ❛ttr✐❜✉t✐♥❣ t♦ t❤❡ ♣r❡♠✐s❡s t❤❡ ❣❧✉t
tr✉t❤✲✈❛❧✉❡ ❛♥❞ t❛❦✐♥❣ t❤❡ ❝♦♥❝❧✉s✐♦♥ t♦ ❜❡ s✐♠♣❧② ❢❛❧s❡✳ ❇② t❤❡ ❞✐❛❧❡t❤❡✐st✬s
❛❝❝♦✉♥t ♦❢ ✈❛❧✐❞✐t②✱ t❤✐s ♠♦✈❡ ❝❛♥♥♦t ❜❡ ♣❡r❢♦r♠❡❞ ✐♥ t❤✐s ❝❛s❡✳ ❍❡r❡✱ t❛❦✐♥❣
t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ❛s t❤❡ ♣r❡♠✐s❡s ♦❢ ❛♥ ❛r❣✉♠❡♥t t♦ tr✐✈✐❛❧✐t②✱
✇❡ ❤❛✈❡ t❤❛t t❤❡② ❛r❡ s✐♠♣❧② ♥♦t ❛ ❣❧✉t✱ s♦✱ ✇❡ ❝❛♥♥♦t r❡♣r♦❞✉❝❡ t❤❡ t②♣✐❝❛❧
❝♦✉♥t❡r✲❡①❛♠♣❧❡ ♦❢ ❣❧✉tt② ♣r❡♠✐ss❡s ❛♥❞ ❢❛❧s❡ ❝♦♥❝❧✉s✐♦♥✱ ❛♥❞ t❤❡ ❞✐❛❧❡t❤❡✐st
♦✇❡s ✉s ❛♥ ❡①♣❧❛♥❛t✐♦♥ ♦♥ ❤♦✇ ❛♥ ✐♥❢❡r❡♥❝❡ t♦ tr✐✈✐❛❧✐t② ✭❡①♣❧♦s✐♦♥✮ ✐s ❜❧♦❝❦❡❞✳
■♥ ♦t❤❡r ✇♦r❞s✿ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✐s ♥♦t r❡❛❧❧② ❝♦✈❡r❡❞ ❜② t❤❡ ❞✐❛❧❡t❤❡✐sts✬ ❧♦❣✐❝✱
✇❡ ❞♦♥✬t ❦♥♦✇ ✇❤❛t ♠❛② ❧❡❣✐t✐♠❛t❡❧② ❜❡ ✐♥❢❡rr❡❞ ❢r♦♠ tr✉t❤ ❛♥❞ ♥♦♥✲tr✉t❤✱
❣✐✈❡♥ t❤❛t t❤❡r❡ ✐s ♥♦ s✉❝❤ tr✉t❤✲✈❛❧✉❡ ❛s ♥♦♥✲tr✉t❤ ✐♥ t❤❡ ✉s✉❛❧ ❞✐❛❧❡t❤❡✐sts✬
❧♦❣✐❝ ✭❛♥❞ ❣✐✈❡♥ t❤❛t ✐t ✐s
♥♦t
❡q✉✐✈❛❧❡♥t t♦ ❢❛❧s❡✱ r❡❝❛❧❧✮✳
❚❤❛t ✐s r❡❛❧❧② ❛
♣r♦❜❧❡♠✱ ❜❡❝❛✉s❡ ❛s ✇❡ ❤❛✈❡ ♠❡♥t✐♦♥❡❞✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✐s
❛♥ ❡①♣❧✐❝✐t ❝♦♥tr❛❞✐❝t✐♦♥✳ ❙♦✱ t❤❡r❡ ✐s ♥♦ ❡①♣❧❛♥❛t✐♦♥ ♦❢ ❤♦✇ ❝❛♥ ✇❡ ❣r❛♥t t❤❛t
❢r♦♠ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ♥♦t ❡✈❡r②t❤✐♥❣ ❢♦❧❧♦✇s✱ ✇❤❡♥ t❤❛t ❝♦♥tr❛❞✐❝t✐♦♥ ✐♥✈♦❧✈❡❞
t❤❡ tr✉t❤ ♣r❡❞✐❝❛t❡ ✐♥ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ❢❛s❤✐♦♥✳
❚❤❛t ❧❛st ♣♦✐♥t ✐s ♠❛❞❡ ❝❧❡❛r❡r ✇❤❡♥ ✇❡ ♥♦t✐❝❡ t❤❛t t❤❡
▲♦❣✐❝ ♦❢ P❛r❛❞♦①
✭▲P✮✱ Pr✐❡st✬s ❢♦r♠❛❧✐③❛t✐♦♥ ♦❢ t❤♦s❡ ✐❞❡❛s✱ ❤❛s s✐♠♣❧② ♥♦ s♣❛❝❡ ❢♦r t❤❡ ❦✐♥❞ ♦❢
❝♦♥tr❛❞✐❝t✐♦♥ ♣r❡s❡♥t ✐♥ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✭s❡❡ Pr✐❡st ❬✺❪✮✳ ❊✈❡r② ❝♦♥tr❛❞✐❝t✐♦♥
❛❧❧♦✇❡❞ ✐♥ ▲P ✐s t❤♦✉❣❤t ♦❢ ❛s r❡♣r❡s❡♥t✐♥❣ ❛ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❛♥❞✱ ❜② t❤❡
❞✐❛❧❡t❤❡✐sts✬ ❧✐❣❤ts✱ ❛❧s♦ ❛ ❣❧✉t✳ ❙♦✱ ❝♦♥tr❛❞✐❝t✐♦♥s t❤❛t ❛r❡ ♥♦t ❣❧✉ts ❛r❡ ❡✐t❤❡r
s✐♠♣❧② ❢❛❧s❡ ♦r ♥♦t r❡♣r❡s❡♥t❡❞✱ ❛♥❞ t❤❛t ❣r❛♥ts t❤❛t t❤❡ ✉s✉❛❧ tr❡❛t♠❡♥t ♦❢
❡①♣❧♦s✐♦♥ ✐♥ ▲P ✇✐❧❧ ♥♦t ❜❡ ❛✈❛✐❧❛❜❧❡ t♦ t❤❡♠✳
P❡r❤❛♣s ❛♥♦t❤❡r tr✉t❤✲✈❛❧✉❡
✭♥♦t✲tr✉❡ ♦r ✉♥tr✉❡✮ ✇♦✉❧❞ ❜❡ ♥❡❡❞❡❞✱ ♦r ♣❡r❤❛♣s✱ t❤❡ ✈❡r② ✐❞❡❛ ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥
✐♥✈♦❧✈❡❞ ❛♥❞ ❝♦♥✢❛t❡❞ ✇✐t❤ ❣❧✉ts ❧❡❛❞s t♦ s✉❝❤ ❡q✉✐✈♦❝❛t✐♦♥ ✭❢♦r ♠♦r❡ ♦♥ s✉❝❤
❞✐st✐♥❝t✐♦♥s ❛♥❞ ❡q✉✐✈♦❝❛t✐♦♥s✱ s❡❡ ❇é③✐❛✉ ❬✸❪✱ ❆r❡♥❤❛rt ❬✶❪✮✳
❖♥❡ ❝♦✉❧❞ tr② t♦ ❜❧♦❝❦ t❤✐s ❜② ❝❧❛✐♠✐♥❣ t❤❛t t❤❡r❡ ✐s ❛ tr❡❛t♠❡♥t ❛✈❛✐❧❛❜❧❡✿
❜② ♣♦✐♥t✐♥❣ t♦ t❤❡ ❢❛❝t t❤❛t
T ✭_L^✮ ∧ ¬T ✭_L^✮ ✐♠♣❧✐❡s T ✭_L^✮ ∧ F ✭_L^✮✱ ✇❡ ❝♦✉❧❞
tr② t♦ ❣r❛♥t t❤❛t ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ✐♠♣❧✐❡s ❛ ❣❧✉t✱ ❛♥❞ ❛ ❣❧✉t ✐s ❛❜❧❡ t♦ ♣r❡✈❡♥t
❡①♣❧♦s✐♦♥✳ ❚❤❛t ✐s✱ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✐s ❛ ❝❛s❡ ♦❢ t❤❡ s✐♠♣❧❡ ▲✐❛r✳ ❍♦✇❡✈❡r✱
t❤❛t ❜❡❣s t❤❡ q✉❡st✐♦♥ ❛❣❛✐♥st ♦✉r ♣♦✐♥t✳ ❲❡ ❝❧❛✐♠ t❤❛t ❛ ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t
✐s ♥♦t ❛ ❣❧✉t ✐s s✐♠♣❧② ♥♦t t♦✉❝❤❡❞ ❜② t❤❡ ❧♦❣✐❝ ♦❢ t❤❡ ❞✐❛❧❡t❤❡✐st✱ s♦✱ ✐t ♠❛②
❧❡❛❞ t♦ tr✐✈✐❛❧✐t② ❛♥❞ ✐♠♣❧② ❡✈❡r②t❤✐♥❣✳ ❚❤❡ ❢❛❝t t❤❛t ✐t ✐♠♣❧✐❡s ❛ ❣❧✉t ✐s ❥✉st
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✾✺
✐rr❡❧❡✈❛♥t ✐♥ t❤✐s ❝❛s❡✳
P❡r❤❛♣s ✇❡ ❝❛♥ ♠❛❦❡ ♦✉r ♣♦✐♥t ❡✈❡♥ ❝❧❡❛r❡r ✐❢ ✇❡ st❛t❡ t❤❡ ♣r♦❜❧❡♠ ✇✐t❤ t❤❡
❡①t❡♥❞❡❞ ▲✐❛r t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ r♦❧❡ ♦❢ ♥❡❣❛t✐♦♥✳ ❈♦♥s✐❞❡r t❤❛t ❛ s❡♥t❡♥❝❡
✐s ♥♦t✲tr✉❡ ♣r❡❝✐s❡❧② ✇❤❡♥ ✐t ❢❛✐❧s t♦ ❜❡ tr✉❡✳ ❚❤✐s ✐s ❛ tr✉✐s♠✱ ❜✉t ✇❤❡♥ ✇❡ ♣❧✉❣
✐t ✐♥ t❤❡ ❞✐❛❧❡t❤❡✐st ♣✐❝t✉r❡ ♦❢ tr✉t❤✲✈❛❧✉❡s✱ t❤❡ r❡s✉❧t ✐s ✐♥t❡r❡st✐♥❣✳ ■t s❡❡♠s
✺ ❛❝ts ❥✉st ❧✐❦❡ t❤❛t✳ ❆s Pr✐❡st ❬✶✵✱
t❤❛t Pr✐❡st ❡①♣❡❝ts t❤❛t ❛ ❣❡♥✉✐♥❡ ♥❡❣❛t✐♦♥
♣✳✼✾❪ ♣✉ts ✐t✿ ✏❬❛❪ ❣❡♥✉✐♥❡ ❝♦♥tr❛❞✐❝t♦r②✲❢♦r♠✐♥❣ ♦♣❡r❛t♦r ✇✐❧❧ ❜❡ ♦♥❡ t❤❛t ✇❤❡♥
❛♣♣❧✐❡❞ t♦ ❛ s❡♥t❡♥❝❡✱
α✱
❝♦✈❡rs ❛❧❧ t❤❡ ❝❛s❡s ✐♥ ✇❤✐❝❤
♦t❤❡r ✇❛② ❛r♦✉♥❞✱ ✐t s❡❡♠s✱
✇❤✐❝❤
α
¬α
α
✐s ♥♦t tr✉❡✑✳ ❖♥ t❤❡
✐s ❢❛❧s❡ ✇❤❡♥ ♥❡❣❛t✐♦♥ ❝♦✈❡rs ❛❧❧ t❤❡ ❝❛s❡s ✐♥
✐s tr✉❡✳ ◆♦✇✱ ❛ s❡♥t❡♥❝❡ ❝❛♥ ❜❡ tr✉❡✱ ❛❝❝♦r❞✐♥❣ t♦ ❞✐❛❧❡t❤❡✐sts✱ ✐♥ t✇♦
❞✐st✐♥❝t ❢❛s❤✐♦♥s✿ ✐t ♠❛② ❜❡ s✐♠♣❧② tr✉❡✱ ♦r tr✉❡ ❛♥❞ ❢❛❧s❡ ✭❛ ❣❧✉t✮✳ ■❢ ♥❡❣❛t✐♦♥
✐s t♦ ❜❡❤❛✈❡ ♣r♦♣❡r❧②✱ ❛s r❡❝♦♠♠❡♥❞❡❞ ❜② Pr✐❡st✱ ❛ s❡♥t❡♥❝❡ t❤❛t ✐s ♥♦t✲tr✉❡
❢❛✐❧s t♦ ❜❡ ❵tr✉❡✬ ✐♥ ❛♥② ♦❢ t❤❡ s❡♥s❡s ♠❡♥t✐♦♥❡❞✳ ■t ✐s ❥✉st ❢❛❧s❡✱ s✐♠♣❧② ❢❛❧s❡✳
1 ✐s tr✉❡✱ 0 ✐s ❢❛❧s❡ ❛♥❞ i ✐s ❣❧✉t✱
1 ♦r i❀ ♥❡❣❛t✐♥❣ s✉❝❤ ❛
✻
s❡♥t❡♥❝❡ ✇✐t❤ ❛ ❣❡♥✉✐♥❡ ♥❡❣❛t✐♦♥ s❤♦✉❧❞ t❤❡♥ ❧❡❛❞ ✉s t♦ 0✳ ■♥ t❤✐s s❡♥s❡✱ ✐❢ t❤❡
P✉t ✐♥ t❡r♠s ♦❢ t❤❡ t❤r❡❡✲✈❛❧✉❡❞ ❧♦❣✐❝ ▲P✱ ✇❤❡r❡
❛♥② s❡♥t❡♥❝❡ t❤❛t ✐s tr✉❡ ❤❛✈❡ ❛s ✐ts tr✉t❤ ✈❛❧✉❡ ❡✐t❤❡r
♥❡❣❛t✐♦♥ ❜❡❤❛✈❡s ♣r♦♣❡r❧②✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✐s ❛ ❝♦♥❥✉♥❝t✐♦♥
❜❡t✇❡❡♥ ❛ s❡♥t❡♥❝❡ t❤❛t r❡❝❡✐✈❡s
1
♦r
i
❛♥❞ ❛ s❡♥t❡♥❝❡ t❤❛t r❡❝❡✐✈❡s
0✳
❚❤❛t ✐s
♥♦t t❤❡ s❝❡♥❛r✐♦ ♣r❡s❡♥t❡❞ ❜② ❞✐❛❧❡t❤❡✐sts ✇❤❡r❡ ❡①♣❧♦s✐♦♥ ❢❛✐❧s❀ r❛t❤❡r✱ t❤❛t ✐s
t❤❡ ❦✐♥❞ ♦❢ s✐t✉❛t✐♦♥ t❤❛t ❧❡❛❞s t♦ ❡①♣❧♦s✐♦♥ ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ ❚❤❡ ❞✐❛❧❡t❤❡✐st
❥✉st ❤❛s ♥♦ tr❡❛t♠❡♥t ❢♦r t❤❛t ❦✐♥❞ ♦❢ s✐t✉❛t✐♦♥✳
❙♦✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ ♦✉r ✜rst ❛r❣✉♠❡♥t ✐s ❝❧❡❛r✿ ✇❤❡♥ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r
✐s t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✱ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❝♦♥tr❛❞✐❝t✐♦♥s✱ ❣❧✉ts✱ ❛♥❞ ❞✐❛❧❡t❤❡✐❛s
❜r❡❛❦s ❞♦✇♥✱ ❛♥❞ t❤❡ ❞✐❛❧❡t❤❡✐st s❡❡♠s t♦ ❤❛✈❡ ♥♦ ❝❧❡❛r ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠✳
❲❤✐❧❡ ✐♥ t❤❡ ❡①t❡♥❞❡❞ ❞❡r✐✈❛t✐♦♥ s❦❡t❝❤❡❞ ❛❜♦✈❡✱ t❤❡ s❡♥t❡♥❝❡
L ♦❢ t❤❡ ❡①t❡♥❞❡❞
▲✐❛r ❡♥❞s ✉♣ ❜❡✐♥❣ tr✉❡ ❛♥❞ ❢❛❧s❡✱ t❤❡ ❞❡r✐✈❛t✐♦♥ ✐ts❡❧❢ ❡❧✐❝✐ts ❛ ❝♦♥❥✉♥❝t✐♦♥
❜❡t✇❡❡♥ tr✉t❤ ❛♥❞ ♥♦♥✲tr✉t❤✱ ✇❤♦s❡ st❛t✉s ✐s ♥♦t ❝♦✈❡r❡❞ ❜② t❤❡ ❞✐❛❧❡t❤❡✐st✳
❖✉r ❞✐❛❣♥♦s✐s ✐s t❤❛t s✉❝❤ ❛ s✐t✉❛t✐♦♥ ❝♦♠❡s ❢r♦♠ ❛ ✉♥✇❛rr❛♥t❡❞ ❛ss♦❝✐❛t✐♦♥
❜❡t✇❡❡♥ t❤❡ t❤r❡❡ ❝♦r❡ ❝♦♥❝❡♣ts ♦❢ ❞✐❛❧❡t❤❡✐s♠✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞
▲✐❛r ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉t ✐t ✐s ♥♦t tr✉❡✱ ❛♥❞ ✐t ✐s ♥♦t ❛ ❣❧✉t✳ ❈♦♥tr❛❞✐❝t✐♦♥s
❛♥❞ ❣❧✉ts ❞♦ ♥♦t ♠❛t❝❤ ✐♥ t❤❡ ❞❡s✐r❡❞ ✇❛②✳ ❆s ✇❡ ♠❡♥t✐♦♥❡❞✱ t❤❡ ❝♦❤❡r❡♥❝❡ ♦❢
t❤❡ ❞✐❛❧❡t❤❡✐st tr❡❛t♠❡♥t t♦ t❤❡ ♣❛r❛❞♦①❡s ✐s q✉❡st✐♦♥❡❞✳
◆♦t✐❝❡ t❤❛t t❤❡ ❞✐❛❧❡t❤❡✐st ❝♦✉❧❞ t❛❦❡ ❛ ❢✉rt❤❡r st❡♣ ❛♥❞ ❝❧❛✐♠ t❤❛t✱ ❤♦✇❡✈❡r
t❤❛t st♦r② ✐s t♦❧❞✱ ✇❡ ✐♥❞❡❡❞ ❤❛✈❡ ❛ ❞✐❛❧❡t❤❡✐❛✿ ❣✐✈❡♥ t❤❛t t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✐s
✭t♦ t❤❡ ❞✐❛❧❡t❤❡✐sts✬ ♠✐♥❞✮ ❛ s♦✉♥❞ ❛r❣✉♠❡♥t✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♠✉st ❜❡ tr✉❡✳ ❙♦✱
✺
❆❝❝♦r❞✐♥❣ t♦ Pr✐❡st ❬✶✵✱ ❝❤❛♣✳✹❪✱ ❛♥② ❛❝❝♦✉♥t ♦❢ ♥❡❣❛t✐♦♥ t❤❛t ✐s ✇♦rt❤ ✐ts ♥❛♠❡ ♠✉st
❣r❛♥t t❤❛t ♥❡❣❛t✐♦♥ ✐s ❛ ❝♦♥tr❛❞✐❝t♦r②✲❢♦r♠✐♥❣ ♦♣❡r❛t♦r t♦ ❜❡❣✐♥ ✇✐t❤✳
❚❤❡ ✉♥❞❡rst❛♥❞✐♥❣
♦❢ ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t ✐s ❛t st❛❦❡ ❤❡r❡ ✐s t❤❡ tr❛❞✐t✐♦♥❛❧ ♦♥❡✱ ❢r♦♠ t❤❡ sq✉❛r❡ ♦❢ ♦♣♣♦s✐t✐♦♥✱
✇❤✐❝❤ ✇❡ s❤❛❧❧ ❡①❛♠✐♥❡ s♦♦♥✿
α
❛♥❞
❜✉t ♥♦t ❜♦t❤✳
✻
❖❢ ❝♦✉rs❡✱ ♥❡❣❛t✐♦♥ ✐♥ ▲P ❞♦❡s
β
♥♦t
❛r❡ ❝♦♥tr❛❞✐❝t♦r✐❡s ✇❤❡♥ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s tr✉❡✱
❞♦ t❤❛t✿ ✐t t❛❦❡s
i ❜❛❝❦ t♦ i❀
t❤❛t ✐s ♦♥❡ ♦❢ t❤❡ r❡❛s♦♥s
s♦♠❡ ❤❛✈❡ ❝❧❛✐♠❡❞ t❤❛t Pr✐❡st✬s ♥❡❣❛t✐♦♥ ✐s ♥♦t ❛ ❣❡♥✉✐♥❡ ❝♦♥tr❛❞✐❝t♦r②✲❢♦r♠✐♥❣ ♦♣❡r❛t♦r✳
✾✻
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
✇❡ ❤❛✈❡ ❛ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥ ✭❛ ❞✐❛❧❡t❤❡✐❛✮✳ ❚❤❛t ✐s✱ t❤❡ ✐❞❡❛ ✐s t❤❛t ❜♦t❤
❛♥❞
¬T ✭_L^✮
T ✭_L^✮
❛r❡ tr✉❡✳ ❚❤✐s s❤♦✇s ❤♦✇❡✈❡r t❤❛t ❞✐❛❧❡t❤❡✐❛s ❛♣♣❡❛r ✐♥ t❤✐s ❝❛s❡
♦♥❧② ❛❢t❡r t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ tr✉t❤ ♣r❡❞✐❝❛t❡ t♦ t❤❡ ✇❤♦❧❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡
❡①t❡♥❞❡❞ ▲✐❛r❀ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ♣❛r❛❞♦①✱ ✐t ✐s ♥♦t t♦ t❤❡
L ❞✐r❡❝t❧② t❤❛t ❡♥❞s ✉♣ ❜❡✐♥❣ ❛ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉t
r❛t❤❡r t❤❡ ❝❧❛✐♠ t❤❛t L ✐s tr✉❡ ❛♥❞ ♥♦t tr✉❡ ✭t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❛r❣✉♠❡♥t✮
t❤❛t ❢♦r♠s ❛ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❛t ✐s✱ L ✐s ♥♦t ✐ts❡❧❢ ❛ ❞✐❛❧❡t❤❡✐❛✱ ❜❡❝❛✉s❡
❡①t❡♥❞❡❞ ▲✐❛r s❡♥t❡♥❝❡
t❤❡ ❝♦♥tr❛❞✐❝t✐♦♥ ❛♣♣❡❛rs ♦♥❧② ✇❤❡♥ t❤❡ tr✉t❤ ♣r❡❞✐❝❛t❡ ✐s ❛❧r❡❛❞② ✐♥✈♦❧✈❡❞ ❛♥❞
♥❡❣❛t✐♦♥ ❛♣♣❧✐❡s ❡①t❡r♥❛❧❧② t♦ t❤❡ tr✉t❤ ♣r❡❞✐❝❛t❡✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❞✐❛❧❡t❤❡✐❛
♠✉st ✐♥✈♦❧✈❡ t❤❡ s❡♥t❡♥❝❡s
tr✉❡✱ ♥♦t
L
T ✭_L^✮
❛♥❞
¬T ✭_L^✮✱
✇❤✐❝❤ ❛r❡ ❜♦t❤ ❞❡❡♠❡❞ t♦ ❜❡
❞✐r❡❝t❧②✳ ❚❤❡ ♦r✐❣✐♥❛❧ s❡♥t❡♥❝❡ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r✱ t❤❡♥✱ ✐s ♥♦t
❛ ❞✐❛❧❡t❤❡✐❛✱ ❝♦♥tr❛r② t♦ ✇❤❛t t❤❡ ❞✐❛❧❡t❤❡✐st ❝❧❛✐♠s✳
s❡♥t❡♥❝❡
L
❙♦ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r
✐s ✐ts❡❧❢ ♥♦t ❛ ❞✐❛❧❡t❤❡✐❛ ❛♥❞ ✇❡ ❛r❡ ❧❡❢t ✇✐t❤ ♥♦ ❛❝❝♦✉♥t ♦❢ ✐ts
st❛t✉s✳ ❙♦✱ t❤✐s ✐s ♥♦t ❛ ❣♦♦❞ ♠♦✈❡✳
❚❤❛t ❧❛st ✇♦rr② ❜r✐♥❣s ✉s ❜❛❝❦ t♦ t❤❡ s✐♠♣❧❡ ▲✐❛r✳ ●✐✈❡♥ t❤❛t t❤❡ ❡①t❡♥❞❡❞
▲✐❛r ♣r❡s❡♥ts ✉s ✇✐t❤ ❛ ❝❛s❡ ♦❢ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t ✐s ♥♦t ❛ ❞✐❛❧❡t❤❡✐❛✱ ♣❡r❤❛♣s
t❤❡r❡ ❛r❡ s♦♠❡ s✉r♣r✐s❡s ✐♥ t❤❡ s✐♠♣❧❡ ▲✐❛r t♦♦❄ ■♥❞❡❡❞ t❤❡r❡ ❛r❡✳ ❈♦♥s✐❞❡r t❤❡
❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ s✐♠♣❧❡ ▲✐❛r✿
T ✭_L^✮ ∧ T ✭_¬L^✮✳
❛ ❞✐❛❧❡t❤❡✐❛✱ ❛ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥✳
❚❤✐s ✐s ✉s✉❛❧❧② ❝❧❛✐♠❡❞ t♦ ❜❡
❍♦✇❡✈❡r✱ ♦♥ ✇❤❛t ❝♦♥❝❡r♥s ❝♦♥tr❛❞✐❝t✐♦♥✱
t❤❡ ❝♦♥❝❧✉s✐♦♥ ❢❛✐❧s t♦ ❤❛✈❡ t❤❡ ❛♣♣r♦♣r✐❛t❡ s②♥t❛❝t✐❝❛❧ ❢♦r♠✦ ■t r❡❛❧❧② ❞♦❡s ♥♦t
❤❛✈❡ t❤❡ ❢♦r♠ ♦❢ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❣✐✈❡♥ t❤❛t t❤❡ ♥❡❣❛t✐♦♥ ♦♣❡r❛t♦r ✐s ✐♥s✐❞❡ t❤❡
tr✉t❤ ♦♣❡r❛t♦r ✭✐t ✐s ❛♥ ✏✐♥t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥✑✱ ❛s Pr✐❡st ❝❛❧❧s ✐t ❬✾✱ ❝❤❛♣✳✹❪✮✳
❚❤✐s ❝♦♥❝❧✉s✐♦♥ ✐s✱ ❛t ❜❡st✱ ❛ st❛t❡♠❡♥t t♦ t❤❡ ❢❛❝t t❤❛t
L
✐s ❛ ❣❧✉t✱ ❜✉t ♥♦t ❛
❝♦♥tr❛❞✐❝t✐♦♥✳ ❙♦✱ ❛s ✇❡ s❤❛❧❧ ❡①♣❧♦r❡ ❢✉rt❤❡r ❧❛tt❡r ✭s❡❡ t❤❡ ♥❡①t t✇♦ s❡❝t✐♦♥s✮✱
❣❧✉ts ❞♦ ♥♦t ♥❡❡❞ t♦ ❝♦♠❡ ❛s ❝♦♥tr❛❞✐❝t✐♦♥s✦ ❲✐t❤♦✉t t❤❡ ❡①❝❧✉s✐♦♥ ♣r✐♥❝✐♣❧❡
✇❡ ❝❛♥♥♦t ❣♦ ❢r♦♠ ❣❧✉ts t♦ ❛ st❛t❡♠❡♥t ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ str✐❝t❧②
s♣❡❛❦✐♥❣✳
❖❢ ❝♦✉rs❡✱ ♦♥❡ ♠❛② ✉s❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥❥✉♥❝t✐♦♥✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❢❛❧✲
T ✭_L^✮ ∧ T ✭_¬L^✮ t♦ ❛
L ∧ ¬L✱ ✇❤✐❝❤ s❛t✐s✜❡s
s✐t② ♠❡♥t✐♦♥❡❞ ❜❡❢♦r❡✱ ❛♥❞ t❤❡ ❚✲s❝❤❡♠❛✱ t♦ ❣♦ ❢r♦♠
❝♦♥tr❛❞✐❝t✐♦♥✳ ■♥ ❢❛❝t✱ ❢r♦♠ t❤❛t ❝♦♥❥✉♥❝t✐♦♥ ✇❡ ❤❛✈❡
t❤❡ r❡q✉✐r❡♠❡♥ts ❧❛✐❞ ❜❡❢♦r❡ t♦ ❜❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❛t ♠❡❛♥s t❤❛t t❤❡ s✐♠♣❧❡
▲✐❛r ✐♠♣❧✐❡s ❛ ❢♦r♠✉❧❛ t❤❛t ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❇✉t ♥♦t✐❝❡ t❤❛t ✐t ✐s ♥♦t ❡q✉✐✈✲
❛❧❡♥t t♦ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ✭t❤❛t ✐s✱ ✐t ❞♦❡s ♥♦t ♠❡❛♥ t❤❡ s❛♠❡ ❛s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱
✐♥ ❛♥② r❡❛s♦♥❛❜❧❡ s❡♥s❡ ♦❢ ❵♠❡❛♥✐♥❣ t❤❡ s❛♠❡✬✮✳
■♥ ❢❛❝t✱ ✇❡ ❝❛♥♥♦t ❣♦ ❢r♦♠
L ∧ ¬L
■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡r❡ ✐s ♥♦
t♦
T ✭_L^✮ ∧ ¬T ✭_L^✮
✇✐t❤♦✉t ❡①❝❧✉s✐♦♥✳
❤♦♠♦♣❤♦♥✐❝ tr❛♥s❧❛t✐♦♥ ❢♦r t❤❡ ❢♦r♠✉❧❛ ✇✐t❤♦✉t tr✉t❤ ♣r❡❞✐❝❛t❡ t♦ ❛ ❢♦r♠✉❧❛
✇✐t❤ tr✉t❤ ♣r❡❞✐❝❛t❡✳ ❲❡ ❝❛♥♥♦t ✐♥s❡rt tr✉t❤ ♣r❡❞✐❝❛t❡s ✐♥ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ❛♥❞
❡♥❞ ✉♣ ✇✐t❤ ❛♥♦t❤❡r ❝♦♥tr❛❞✐❝t✐♦♥✳
❙♦✱ ❡①t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥s ❞♦ ♥♦t ♠❡❛♥
t❤❡ s❛♠❡ ❛s ❣❧✉ts✿ ♦♥❡ ❝❛♥♥♦t s❧✐❞❡ ❜❡t✇❡❡♥ t❤❡♠ t❤r♦✉❣❤ ❛ ❝❤❛✐♥ ♦❢ ❡q✉✐✈❛✲
❧❡♥❝❡s✳ ❚❤❛t ✐s ❡♥♦✉❣❤✱ ✐t s❡❡♠s t♦ ✉s✱ t♦ ❣r❛♥t t❤❛t t❤❡ ♥♦t✐♦♥ ♦❢ ❛♥ ❡①t❡r♥❛❧
❝♦♥tr❛❞✐❝t✐♦♥ ❛♥❞ ❣❧✉ts ❛r❡ ♥♦t t❤❡ s❛♠❡✳ ❚❤❡ s✐♠♣❧❡ ▲✐❛r ✭❛ ❣❧✉t✮ ♠❛② ✐♠♣❧②
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✾✼
❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉t ♦♥❡ ❝❛♥♥♦t ❣♦ ❢r♦♠ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ t♦ ❛ ❣❧✉t✳
❖❢ ❝♦✉rs❡✱ ✇❡ ❝♦✉❧❞ ❞❡✜♥❡ ❛ ♣❛✐r ♦❢ ❝♦♥tr❛❞✐❝t♦r✐❡s ✐♥ s❡♠❛♥t✐❝ t❡r♠s✱ ❜✉t
t❤❛t ❞❡✜♥✐t✐♦♥ ✐s s✉❝❤ t❤❛t ✐t ❞♦❡s ♥♦t ✇♦r❦ t♦ t❤❡ ❜❡♥❡✜t ♦❢ t❤❡ ❞✐❛❧❡t❤❡✐st✿
tr❛❞✐t✐♦♥❛❧❧②✱ ❛ ♣❛✐r ♦❢ s❡♥t❡♥❝❡s α ❛♥❞ β ❛r❡ ❝♦♥tr❛❞✐❝t♦r② ✭✐♥ s❡♠❛♥t✐❝ t❡r♠s✮
✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡② ❝❛♥♥♦t ❜♦t❤ ❜❡ tr✉❡✱ ❛♥❞ t❤❡② ❝❛♥♥♦t ❜♦t❤ ❜❡ ❢❛❧s❡✳ ❚❤❛t
✐s✱ t❤❡② ❡❛❝❤ ❤❛✈❡ ♦♥❡ ♦❢ t❤❡ t✇♦ tr✉t❤ ✈❛❧✉❡s✱ ❛♥❞ ♦♥❧② ♦♥❡ ✭Pr✐❡st ❛❧s♦ r❡❝✲
♦❣♥✐③❡s t❤❛t ❞❡✜♥✐t✐♦♥ ❛s ✐♥t✉✐t✐✈❡❧② ❝♦rr❡❝t ✐♥ Pr✐❡st ❬✶✵✱ ♣✳✼✽❪✮✳ ❚❤❡ s❡♠❛♥t✐❝
❞❡✜♥✐t✐♦♥ ✇♦✉❧❞ ❢♦r❜✐❞ ❣❧✉ts✱ ❛♥❞ ✇♦✉❧❞ ❜❡ ✐♥❛❞❡q✉❛t❡ ❢♦r t❤❡ ❞✐❛❧❡t❤❡✐st ✭s❡❡
❛❧s♦ ❬✸❪✱ ❬✶❪✮✳ ❖❢ ❝♦✉rs❡✱ t❤❡ ❞✐❛❧❡t❤❡✐st ♠❛② ❝❤❛♥❣❡ t❤❡ ❝❧❛ss✐❝❛❧ t❡r♠✐♥♦❧♦❣②
t♦ ♠❛❦❡ ❤❡r ❝❛s❡✱ ❜✉t t❤❡♥ t❤❡ ❞✐s♣✉t❡ ✇✐t❤ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✐❛♥s ✇✐❧❧ ❜❡ ♠❡r❡❧② ❛
t❡r♠✐♥♦❧♦❣✐❝❛❧ ❞✐s♣✉t❡✱ s♦♠❡t❤✐♥❣ ❡✈❡♥ ❞✐❛❧❡t❤❡✐sts ✇✐s❤ t♦ ❞❡♥②✳
❚♦ ❦❡❡♣ ✇✐t❤ t❤❡ s❡♠❛♥t✐❝ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥tr❛❞✐❝t♦r✐❡s✱ Pr✐❡st t②♣✐❝❛❧❧②
❜❡❧✐❡✈❡s t❤❛t ♦♥❡ ♠❛② ❤❛✈❡ ❛ ♥❡❣❛t✐♦♥ s✐❣♥ ¬ r❡♣r❡s❡♥t✐♥❣ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✲
❢♦r♠✐♥❣ ♦♣❡r❛t♦r ✭s♦ t❤❛t α ❛♥❞ ¬α ❛r❡ ❝♦♥tr❛❞✐❝t♦r✐❡s ✐♥ t❤❡ ❛❜♦✈❡ s❡♠❛♥t✐❝
s❡♥s❡✮ ❛♥❞ ❛❧s♦ t❤❛t ♦♥❡ ♠❛② ❤❛✈❡ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ✭s♦ t❤❛t α ❛♥❞ ¬α ♠❛②
❜♦t❤ ❜❡ tr✉❡❀ s❡❡ Pr✐❡st ❬✶✵✱ ❝❤❛♣✳✹❪✮✳ ❍♦✇❡✈❡r✱ t❤✐s ♠♦✈❡ s✐♠♣❧② ✐❧❧✉str❛t❡s t❤❡
t②♣✐❝❛❧ ❛♠❜✐❣✉✐t② ✐♥ ❞✐❛❧❡t❤❡✐s♠ t❤❛t ✇❡ ❛r❡ ❜r✐♥❣✐♥❣ t♦ ❧✐❣❤t ❤❡r❡✳ ■❢ α ❛♥❞ ¬α
❛r❡ t♦ ❜❡ ❝♦♥tr❛❞✐❝t♦r✐❡s ✐♥ t❤❡ s❡♠❛♥t✐❝ s❡♥s❡✱ t❤❡♥✱ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ♠✉st
❜❡ tr✉❡✱ ❜✉t ♥♦t ❜♦t❤✳ Pr✐❡st ❛r❣✉❡s t❤❛t ✇❡ ♠❛② ❤❛✈❡ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ✭♥♦✇
❝♦♥tr❛❞✐❝t✐♦♥s ❛r❡ ✉♥❞❡rst♦♦❞ ❛s ❛ ♣❛✐r ♦❢ ❢♦r♠✉❧❛s α ❛♥❞ ¬α✱ ✐♥ t❤❡ s❡♥s❡ ♦❢
❛ ♣❛✐r ♦❢ ❢♦r♠✉❧❛s ♦❢ ❛ ❣✐✈❡♥ s②♥t❛❝t✐❝❛❧ ❢♦r♠✮ ✐❢ ✇❡ ❛❧❧♦✇ t❤❛t s♦♠❡ s❡♥t❡♥❝❡s
♠❛② ❜❡ tr✉❡ ❛♥❞ ❢❛❧s❡✱ ✐✳❡✳✱ ❣❧✉ts✳ ❖❢ ❝♦✉rs❡✱ ✐❢ α ✐s ❛ ❣❧✉t✱ ❜② ❞❡✜♥✐t✐♦♥ ♦❢
❣❧✉t ❛♥❞ ♦❢ ❢❛❧s✐t②✱ ✇❡ ❤❛✈❡ t❤❛t α ✐s tr✉❡ ❛♥❞ ¬α ✐s ❛❧s♦ tr✉❡✳ ❇✉t t❤✐s s✐♠♣❧②
✈✐♦❧❛t❡s t❤❡ r❡q✉✐r❡♠❡♥t ♦❢ t❤❡ s❡♠❛♥t✐❝ ♥♦t✐♦♥ ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥✿ t❤❛t ✇❡ ❤❛✈❡
♦♥❡ ♦❢ t❤❡♠✱ ❜✉t ♥♦t ❜♦t❤✳ ●❧✉ts ❛r❡ ❝❧❡❛r❧② ❢♦r❜✐❞❞❡♥ ❜② t❤❛t ❞❡✜♥✐t✐♦♥✳ Pr✐❡st
✐♥tr♦❞✉❝❡s t❤❡ s❡♠❛♥t✐❝ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉t t❤❡♥✱ ✇❤❡♥ ✐t ❝♦♠❡s t♦
❞❡❢❡♥❞ t❤❛t s♦♠❡ ❝♦♥tr❛❞✐❝t✐♦♥s ♠❛② ❜❡ tr✉❡✱ ✐t ✐s t❤❡ s②♥t❛❝t✐❝❛❧ ♥♦t✐♦♥ t❤❛t
❞♦❡s t❤❡ ❥♦❜✳ ❚❤❡ s❡♠❛♥t✐❝ ❞❡✜♥✐t✐♦♥✱ ❛s ✇❡ ❤❛✈❡ ❝❧❛✐♠❡❞✱ ✐s ✈✐♦❧❛t❡❞✳ ❙♦✱
❛ s❡♠❛♥t✐❝ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥✱ t❤❡ ♦♥❡ ❛✉t❤♦r✐③❡❞ ❡✈❡♥ ❜② Pr✐❡st✱ ✐s
✐♥❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❣❧✉ts ❛♥❞ ❞✐❛❧❡t❤❡✐❛s✳
❙♦✱ ♦✈❡r❛❧❧✱ t❤❡ ▲✐❛r ❞♦❡s ♥♦t s❡❡♠ t♦ ❝♦rr♦❜♦r❛t❡ t❤❡ st♦r② t♦❧❞ ❜② t❤❡
❞✐❛❧❡t❤❡✐st✳ ❊✐t❤❡r ✐t ♣♦s❡s ✉s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉t ♥❡✐t❤❡r ❞✐❛❧❡t❤❡✐❛ ♥♦r ❣❧✉t✱
♦r ❡❧s❡ ✐t ❛❞✈❛♥❝❡s ❛ ❞✐❛❧❡t❤❡✐❛✱ ❜✉t t❤❡ ❧❛❝❦ ♦❢ t❤❡ ❡①❝❧✉s✐♦♥ ♣r✐♥❝✐♣❧❡ ♣r❡✈❡♥ts
✐t ❢r♦♠ ❜❡✐♥❣ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❘❡❝❛❧❧ ❛❧s♦ t❤❛t ✐♥ t❤❡ ✜rst ❝❛s❡ t❤❡r❡ ✐s ♥♦
♣r♦♣❡r❧② ♣❛r❛❝♦♥s✐st❡♥t tr❡❛t♠❡♥t ❢♦r ❛ ❝♦♥tr❛❞✐❝t✐♦♥✿ ♥♦t ❜❡✐♥❣ ❛ ❣❧✉t✱ ✇❡
❤❛✈❡ ♥♦ ❡①♣❧❛♥❛t✐♦♥ ♦❢ ❤♦✇ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ❞♦❡s ♥♦t ❧❡❛❞ ✉s t♦ tr✐✈✐❛❧✐t②✳ ■♥
❢❛❝t✱ r❡❝❛❧❧ t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❝♦♥❥✉♥❝t✐♦♥ ❜❡t✇❡❡♥ tr✉❡ ❛♥❞
♥♦t✲tr✉❡✱ ✇❤✐❝❤ ✐s ❧❡❢t ✉♥❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❞✐❛❧❡t❤❡✐sts✬ ❝❛♥♦♥✳
✾✽
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
✸ ❯♥❞❡rst❛♥❞✐♥❣ ❣❧✉ts
P❡r❤❛♣s s♦♠❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❞✐✣❝✉❧t✐❡s ❝♦✉❧❞ ❜❡ s♦❧✈❡❞ ❜② ❛❞♦♣t✐♥❣ t❤❡ ❡①✲
❝❧✉s✐♦♥ ♣r✐♥❝✐♣❧❡✳ ■♥ t❤❛t ❝❛s❡✱ ✇❡ ✇♦✉❧❞ ❤❛✈❡ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ T ✭_¬L^✮
❛♥❞ ¬T ✭_L^✮✳ ❲❤② ❡①❛❝t❧② ❞♦❡s♥✬t Pr✐❡st ❛❞♦♣t ❡①❝❧✉s✐♦♥❄ ❊❛s②✿ r❡❝❛❧❧ t❤❛t
✐♥t✉✐t✐✈❡❧②✱ ❡①❝❧✉s✐♦♥ ♣r❡✈❡♥ts t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❣❧✉ts✳ ❍♦✇❡✈❡r✱ ❛s ✇❡ ❛r❣✉❡❞
✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✐s ♥♦t ❛ ❣❧✉t✱ ❜✉t
✐t ✐s ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❙♦✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r✱ t❤❡r❡ ✇♦✉❧❞ ♥♦t ❜❡ ❛
♣r♦❜❧❡♠✱ ❛❞♦♣t✐♥❣ ❡①❝❧✉s✐♦♥ ✇♦✉❧❞ r✉❧❡ ♦✉t s♦♠❡t❤✐♥❣ ✇❡ ❛❧r❡❛❞② ❞♦ ♥♦t ❤❛✈❡✳
❇✉t ❧❡t ✉s ♥♦t ❣♦ s♦ ❢❛st✳
❚❤❡ ♠❛✐♥ ❛r❣✉♠❡♥ts ❛❣❛✐♥st ❡①❝❧✉s✐♦♥ ❛r❡ s✐♠♣❧❡ ✭Pr✐❡st ❬✾✱ ❝❤❛♣✳✹❪✮✿ ✶✮
t❤❡r❡ ❛r❡ ♥♦ ❣♦♦❞ r❡❛s♦♥s t♦ ❛❞♦♣t ✐t❀ ✷✮ ❡①❝❧✉s✐♦♥ ✇♦✉❧❞ s♣r❡❛❞ ✐♥❝♦♥s✐st❡♥❝✐❡s✱
t❤❛t ✐s✱ ❛♥② st❛t❡♠❡♥t ♦❢ t❤❡ ❢♦r♠ T ✭_α^✮ ∧ T ✭_¬α^✮ ❛❧s♦ ❜❡❝♦♠❡s ❛ ❝♦♥tr❛❞✐❝t✐♦♥
♦❢ t❤❡ ❢♦r♠ T ✭_α^✮ ∧ ¬T ✭_α^✮✳ ❚❤❛t ✐s✱ ❛♥② ✐♥t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥ ✇♦✉❧❞ ❜❡ ❛❧s♦
❛♥ ❡①t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥✱ t❤✉s ♠✉❧t✐♣❧②✐♥❣ ❝♦♥tr❛❞✐❝t✐♦♥s ❜❡②♦♥❞ ♥❡❝❡ss✐t②✳
◆♦✇✱ ❧❡❛✈✐♥❣ ❛s✐❞❡ t❤❡ ❢❛❝t t❤❛t ✐t ✐s str❛♥❣❡ t♦ ❤❛✈❡ t✇♦ ❞✐st✐♥❝t ❦✐♥❞s ♦❢
❝♦♥tr❛❞✐❝t✐♦♥s✱ ✐♥t❡r♥❛❧ ❛♥❞ ❡①t❡r♥❛❧ ✭s❡❡ ❛❧s♦ ❬✶❪✮✱ t❤❡ s❡❝♦♥❞ ♣♦✐♥t ✐s ❣♦✐♥❣
t♦ ❜❡ ❝r✉❝✐❛❧ ❤❡r❡✳ ❋♦r t❤❡ s❛❦❡ ♦❢ ❛r❣✉♠❡♥t✱ ❧❡t ✉s ❝♦♥❝❡❞❡ t❤❛t ✐♥t❡r♥❛❧ ❝♦♥✲
tr❛❞✐❝t✐♦♥s ❛r❡ r❡❛❧❧② ❝♦♥tr❛❞✐❝t✐♦♥s ❥✉st ❛s ❡①t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥s ✭r❡❝❛❧❧ t❤❛t
✇❤❛t Pr✐❡st ❝❛❧❧s ✐♥t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥s ❞♦ ♥♦t ❤❛✈❡ t❤❡ s②♥t❛❝t✐❝❛❧ ❢♦r♠ ♦❢ ❛
❝♦♥tr❛❞✐❝t✐♦♥✮✳ Pr✐❡st ✇✐s❤❡s t♦ ❦❡❡♣ ❞✐❛❧❡t❤❡✐❛s t♦ ❛ ♠✐♥✐♠✉♠✳ ■t ✐s ♥♦t t❤❡
❝❛s❡ t❤❛t ❡✈❡r② ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t ♦♥❡ ❢❛❝❡s s❤♦✉❧❞ ❜❡ ❛❝❝❡♣t❡❞ ❛♥❞ t❛❦❡♥ ❛s
tr✉❡✳ ❉✐❛❧❡t❤❡✐❛s ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡ r❛r❡✱ ❡✈❡♥ ❢♦r ❛ ❞✐❛❧❡t❤❡✐st ✭✉♥❧❡ss ♦♥❡
✐s ❛❧s♦ ❛ tr✐✈✐❛❧✐st❀ s❡❡ ❬✹❪ ❢♦r ❢✉rt❤❡r ❞✐s❝✉ss✐♦♥✮✳ ❚❤❛t ♣r✐♥❝✐♣❧❡ ✐s ❜❛❝❦❡❞ ❜②
t✇♦ ❦✐♥❞s ♦❢ ❝❧❛✐♠s✳ ❚❤❡ ✜rst ❝❧❛✐♠ ❝♦♥❝❡r♥s ❡❝♦♥♦♠②✱ ❛♥❞ ❛♣♣❡❛rs ✐♥ t❤❡
❢♦r♠ ♦❢ Pr✐❡st✬s r❛③♦r ❬✾✱ ♣✳✶✶✻❪✿ ❞♦ ♥♦t ♠✉❧t✐♣❧② ❝♦♥tr❛❞✐❝t✐♦♥s ❜❡②♦♥❞ ♥❡❝❡s✲
s✐t②✳ ❊①❝❧✉s✐♦♥ ♠❛❦❡s ❛♥ ✐♥t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥ ✐♥t♦ ❛♥ ❡①t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥✱
s♦ ❝♦♥tr❛❞✐❝t✐♦♥s s♣r❡❛❞ ✇✐t❤♦✉t ♥❡❝❡ss✐t②✳ ❚❤❡ s❡❝♦♥❞ r❡❛s♦♥ ❝♦♥❝❡r♥s t❤❡
✈❡r② ✉s❡❢✉❧♥❡ss ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥s✿ ✇❤❡♥❡✈❡r t❤❡r❡ ✐s ❛♥ ❡①♣❧❛♥❛t✐♦♥ ❢♦r ❛ ♣❤❡✲
♥♦♠❡♥♦♥ t❤❛t ❞♦❡s ♥♦t r❡q✉✐r❡ t❤❡ ♣♦st✉❧❛t✐♦♥ ♦❢ ❛ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥✱ t❤❡♥✱
♦t❤❡r ❝♦♥s✐❞❡r❛t✐♦♥s ❜❡✐♥❣ ❡q✉❛❧✱ ✇❡ s❤♦✉❧❞ ♣r❡❢❡r t❤❡ ❝♦♥s✐st❡♥t ❡①♣❧❛♥❛t✐♦♥
✭s❡❡ ❬✽❪✮✳ ❆❝❝♦r❞✐♥❣ t♦ Pr✐❡st✱ ▲✐❛r ♣❛r❛❞♦①❡s ❛r❡ ❝❛s❡s t❤❛t ❝❛♥♥♦t ❜❡ ❛❞❡✲
q✉❛t❡❧② tr❡❛t❡❞ ❜② ❝♦♥s✐st❡♥t t❤❡♦r✐❡s✱ s♦ t❤✐s ✐s ❛ ❝❛s❡ ✐♥ ✇❤✐❝❤ ❛❞♦♣t✐♦♥ ♦❢
tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ✐s ❥✉st✐✜❡❞✳
❚❤❡♥✱ t❤❡ r❡❛s♦♥ t♦ ❛✈♦✐❞ ❡①❝❧✉s✐♦♥ ✐s r❡❧❛t❡❞ t♦ t❤❡ ❝♦♠♠❛♥❞ t♦ ❦❡❡♣
❞✐❛❧❡t❤❡✐❛s t♦ ❛ ♠✐♥✐♠✉♠✳ ❍♦✇❡✈❡r✱ t❤❡ ❝♦♥✢❛t✐♦♥ ❜❡t✇❡❡♥ ❞✐❛❧❡t❤❡✐❛s ❛♥❞
❣❧✉ts ♠❛❦❡s t❤❛t ❦✐♥❞ ♦❢ ❝♦♥tr♦❧ ✐♠♣♦ss✐❜❧❡✳ ■♥❞❡❡❞✱ ❧❡t ✉s ❛ss✉♠❡ t❤❛t ✇❡ ❤❛✈❡
❛ ❞✐❛❧❡t❤❡✐❛✱ ♣❡r❤❛♣s t❤❡ s✐♠♣❧❡ ▲✐❛r s❡♥t❡♥❝❡ L✳ ❚❤❡♥✱ ❜② ❞❡✜♥✐t✐♦♥✱ T ✭_L^✮
❛♥❞ F ✭_L^✮✳ ❈❛♥ ✇❡ ❝♦♥✜♥❡ ❣❧✉ts ♦♥❧② t♦ t❤❛t ♦♥❡ s❡♥t❡♥❝❡❄ ◆♦t ❛❝❝♦r❞✐♥❣ t♦
t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❣❧✉ts ✐♥ t❤❡ ❧♦❣✐❝ ▲P✦ ▲❡t ✉s ❝❤❡❝❦✳
▲❡t ✉s t❛❦❡ L ❛s ♦✉r ❣❧✉t✳ ❲❤❛t ❝♦✉❧❞ ✇❡ s❛② ❛❜♦✉t ¬L❄ ❲❡❧❧✱ ❛❝❝♦r❞✐♥❣
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✾✾
t♦ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ♥❡❣❛t✐♦♥ ✐♥ ▲P✱ t❤❡ ♥❡❣❛t✐♦♥ ♦❢ ❛ ❣❧✉t ✐s ❛❧✇❛②s ❛ ❣❧✉t t♦♦✳
❙♦✱ ❣❧✉ts ❛❧✇❛②s ❝♦♠❡ ✐♥ ♣❛✐rs✳ ❚❤❛t ✐s ♥♦t ❛❧❧✱ ❤♦✇❡✈❡r✳ ❚❤❡ ❝♦♥❥✉♥❝t✐♦♥ ♦❢
❛ ❣❧✉t✱ ❧✐❦❡ t❤❡ s✐♠♣❧❡ ▲✐❛r
L✱
✇✐t❤ ❛♥② tr✉❡ s❡♥t❡♥❝❡
✐s ❛❧✇❛②s ❛ ❣❧✉t t♦♦✱ s♦ t❤❛t ✏ S
∧ L✑
S✱
❧✐❦❡ ✏s♥♦✇ ✐s ✇❤✐t❡✑✱
✐s ❛♥♦t❤❡r ❣❧✉t✱ ❥✉st ❛s ✏ S
∧ ¬L✑✳
♥❡❣❛t✐♦♥s ♦❢ t❤♦s❡ ❦✐♥❞s ♦❢ s❡♥t❡♥❝❡s✱ ❛s ✇❡ ♠❡♥t✐♦♥❡❞✱ ❛r❡ ❛❧s♦ ❣❧✉ts✳
❚❤❡
❚❤❛t
❛❧r❡❛❞② ❛❝❝♦✉♥ts ❢♦r q✉✐t❡ ❛ ❧♦t ♦❢ ❣❧✉ts✱ ✇❤❡♥ t❤❡ ♣r♦❝❡ss❡s ❛r❡ ❢✉rt❤❡r ✐t❡r❛t❡❞✳
❇✉t t❤❡r❡ ✐s ♠♦r❡✳
❚❤❡ ❞✐s❥✉♥❝t✐♦♥ ♦❢ ❛ ❣❧✉t ✇✐t❤ ❛♥② ❢❛❧s❡ s❡♥t❡♥❝❡
G✱
❧✐❦❡
✏s♥♦✇ ✐s ❣r❡❡♥✑✱ ✐s ❛ ❣❧✉t t♦♦❀ t❤❛t ✐s✱ ✏ L ∨ G✑ ✐s ❛ ❣❧✉t✱ ❥✉st ❛s ✏ ¬L ∨ G✑✳ ❚❤❡✐r
♥❡❣❛t✐♦♥s ❛r❡ ❣❧✉ts t♦♦✱ ❛♥❞ s♦ ♦♥✳
❙♦✱ ❣❧✉ts ♠✉❧t✐♣❧② ✈❡r② ❢❛st❀ ♦♥❡ ❝❛♥♥♦t
❦❡❡♣ ❝♦♥tr♦❧ ♦✈❡r t❤❡ ✐♥❢❡st❛t✐♦♥✿ ❢❡❡❞ t❤❡ ❧♦❣✐❝ ▲P ✇✐t❤ ❛ s✐♥❣❧❡ ❣❧✉t ❛♥❞ ②♦✉
s❤❛❧❧ ❡❛s✐❧② ❣♦ ♠✉❝❤ ❜❡②♦♥❞ ♥❡❝❡ss✐t②✳
❙♦✱ ✐❢ ❣❧✉ts ❛r❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ✭❛♥❞ ✇❡ ❛r❡ s✉♣♣♦s✐♥❣
t❤✐s✱ ❢♦r t❤❡ s❛❦❡ ♦❢ ❛r❣✉♠❡♥t✮✱ t❤❡r❡ ✇✐❧❧ ❜❡ ♠❛♥② tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s✱ ♠✉❝❤
♠♦r❡ t❤❛♥ ❛ ♠♦❞❡st ❞✐❛❧❡t❤❡✐st ✇♦✉❧❞ ❧✐❦❡ t♦ ❛❞♠✐t✳ ●✐✈❡♥ ♦♥❡ s✐♥❣❧❡ s❡♥t❡♥❝❡
t❤❛t ❤❛s ❛ ❣❧✉t ❛s ❛ tr✉t❤ ✈❛❧✉❡ ✭❛♥❞ t❤❡ ❞✐❛❧❡t❤❡✐st ❛❞♠✐ts t❤❛t t❤❡ s✐♠♣❧❡ ▲✐❛r
✐s ♦♥❡ s✉❝❤✮✱ ♣r❛❝t✐❝❛❧❧② ❡✈❡r② tr✉❡ ❛♥❞ ❡✈❡r② ❢❛❧s❡ s❡♥t❡♥❝❡ ✇✐❧❧ ❣❡t ✐♥✈♦❧✈❡❞
✐♥ ❛ ❝♦♠♣♦s❡❞ s❡♥t❡♥❝❡ t❤❛t ✐s ❛❧s♦ ❛ttr✐❜✉t❡❞ ❛ ❣❧✉t tr✉t❤ ✈❛❧✉❡✳ ❚❤❛t ❝❧❡❛r❧②
❛tt❡sts ❛❣❛✐♥st t❤❡ ✐❞❡❛ t❤❛t ❞✐❛❧❡t❤❡✐❛s ❛r❡ r❛r❡ ❛♥❞ s❤♦✉❧❞ ♥♦t ❜❡ ♠✉❧t✐♣❧✐❡❞✱
❜r✐♥❣✐♥❣ ❞✐❛❧❡t❤❡✐s♠ ❝❧♦s❡r t♦ ❛ tr✐✈✐❛❧ ♣♦s✐t✐♦♥✳
❲❡ ❞♦ ♥♦t ✐♠♣✉t❡ ❛ ✈❡rs✐♦♥ ♦❢ tr✐✈✐❛❧ ❞✐❛❧❡t❤❡✐s♠ t♦ Pr✐❡st ❛♥❞ ♦t❤❡r ❞✐✲
✼ ❚❤❡② ♠❛② ❝❧❡❛r❧② ❤♦❧❞ t❤❛t s❡♥t❡♥❝❡s s✉❝❤ ❛s ✏s♥♦✇ ✐s ✇❤✐t❡✑ ❛♥❞
❛❧❡t❤❡✐sts✳
✏✷ ✰ ✷ ❂ ✹✑ ❛r❡ ❥✉st tr✉❡ ✭❛❧t❤♦✉❣❤ t❤❡② ✇✐❧❧ ❤❛✈❡ tr♦✉❜❧❡ t♦ ❡①♣r❡ss ✐t✮✱ ❛♥❞
♦t❤❡rs s✉❝❤ ❛s ✏❙♥♦✇ ✐s r❡❞✑ ❛r❡ ♦♥❧② ❢❛❧s❡ ✭✐❞❡♠✮✳ ❖✉r ❝❧❛✐♠ ✐s s✐♠♣❧❡r✿ ❣✐✈❡♥
❛ s✐♥❣❧❡ ❣❧✉t✱ ✐t ✐s ✈✐rt✉❛❧❧② ✐♠♣♦ss✐❜❧❡ t♦ ❛✈♦✐❞ ❛ ❞❡♠♦❣r❛♣❤✐❝ ❡①♣❧♦s✐♦♥ ✐♥ t❤❡
♣♦♣✉❧❛t✐♦♥ ♦❢ ❣❧✉ts ✐❢ ▲P ✐s ❛❞♠✐tt❡❞ ❛s t❤❡ ✉♥❞❡r❧②✐♥❣ ❧♦❣✐❝✳ ❙♦ ✇❤✐❧❡ t❤❛t ✐s
♥♦t q✉✐t❡ ❛ tr✐✈✐❛❧ ❞✐❛❧❡t❤❡✐s♠✱ ✐t ✐s ❝❧♦s❡ ❡♥♦✉❣❤ t♦ ❝❛✉s❡ tr♦✉❜❧❡ t♦ Pr✐❡st✬s
r❛③♦r✿ ❞✐❛❧❡t❤❡✐❛s ❛r❡ ✐♥❞❡❡❞ ❛❜✉♥❞❛♥t✳
❆♥❞✱ ❣✐✈❡♥ ♦✉r ❛ss✉♠♣t✐♦♥ t❤❛t ❞✐✲
❛❧❡t❤❡✐❛s ❛r❡ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s✱ t❤❡r❡ ❛r❡ ❥✉st ❛s ♠✉❝❤ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛s
t❤❡r❡ ❛r❡ ❢♦r♠✉❧❛s✳
◆♦t✐❝❡ t❤❛t ♠✉❝❤ t❤❡ s❛♠❡ ❤❛♣♣❡♥s ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ ●✐✈❡♥ ❛ tr✉❡ ❢♦r♠✉❧❛✱
♦♥❡ ❝❛♥ ♣r♦❞✉❝❡ ❧♦ts ♦❢ ♦t❤❡r tr✉❡ ❢♦r♠✉❧❛s ❜② ❡♠♣❧♦②✐♥❣ t❤❡ ❝♦♥♥❡❝t✐✈❡s✳ ❙♦✱
✐s t❤❡r❡ ❛ s♣❡❝✐❛❧ ♣r♦❜❧❡♠ ❢♦r ❣❧✉ts❄ ❖❢ ❝♦✉rs❡✦ ❲❤❡♥ ✇❡ ✐❞❡♥t✐❢② ❣❧✉ts ✇✐t❤
tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s✱ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ✇✐❧❧ ♠✉❧t✐♣❧② ❜❡②♦♥❞ ♥❡❝❡ss✐t②✳ ❚r✉t❤s
♠❛② ♠✉❧t✐♣❧②❀ ✇❡ ✉s✉❛❧❧② t❤✐♥❦ ✐t ❛ ♥✐❝❡ ❢❛❝t t❤❛t ✇❡ ❝❛♥ ❣❡t ❢✉rt❤❡r tr✉t❤s
❢r♦♠ ♦♥❡ tr✉t❤✳ ❍♦✇❡✈❡r✱ ❝♦♥tr❛❞✐❝t✐♦♥s s❤♦✉❧❞ ♥♦t ♠✉❧t✐♣❧②❀ t❤❡② s❤♦✉❧❞ ❜❡
t❤❡ ❡①❝❡♣t✐♦♥✱ ❡✈❡♥ ✐♥ t❤❡ ❞✐❛❧❡t❤❡✐st ❢r❛♠❡✇♦r❦❀ ❛❧❛s✱ t❤❡② ❛r❡ ♥♦t✳ ❚❤❡r❡ ✐s ❛
✼
❇é③✐❛✉ ❬✹❪ ❛r❣✉❡s t❤❛t Pr✐❡st ❛♥❞ ❞❡❢❡♥❞❡rs ♦❢ ▲P ❝❛♥♥♦t ❛✈♦✐❞ tr✐✈✐❛❧ ❞✐❛❧❡t❤❡✐s♠✳ ●✐✈❡♥
t❤❛t ✐♥ ▲P ✐t ✐s ♣♦ss✐❜❧❡ ❢♦r ❡✈❡r② ❢♦r♠✉❧❛ t♦ ❜❡ ❛ ❣❧✉t✱ ❤❡ ❛r❣✉❡s✱ t❤❡♥ ❡✈❡r② ❢♦r♠✉❧❛ ✐s ✐♥❞❡❡❞
❛ ❣❧✉t✳ ❇② t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❣❧✉ts ❛♥❞ ❝♦♥tr❛❞✐❝t✐♦♥s✱ tr✐✈✐❛❧✐s♠ r❡s✉❧ts✳ ❲❡ ❞♦ ♥♦t ❣♦ s♦
❢❛r✱ ❛❧t❤♦✉❣❤ ✇❡ t❤✐♥❦ ♦✉r ❛r❣✉♠❡♥t ❢♦r❝❡s t❤❡ ❞✐❛❧❡t❤❡✐st t♦ r❡❝♦❣♥✐③❡ ♠✉❝❤ ♠♦r❡ ❣❧✉ts t❤❛♥
s❤❡ ❡①♣❡❝t❡❞ ✭✇❤✐❝❤ ✐s ❛❧r❡❛❞② ❜❛❞ ❡♥♦✉❣❤✮✳
✶✵✵
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
❜✐❣ ❞✐✛❡r❡♥❝❡ ✐♥ tr✉t❤s ♠✉❧t✐♣❧②✐♥❣ t❤❡♠s❡❧✈❡s ❛♥❞ ❝♦♥tr❛❞✐❝t✐♦♥s ♠✉❧t✐♣❧②✐♥❣
t❤❡♠s❡❧✈❡s✳ ❚❤❡r❡ ✐s ♥♦ s♣❡❝✐❛❧ r❡❛s♦♥ t♦ ❜❡❧✐❡✈❡ ✐♥ t❤❡ tr✉t❤ ♦❢ t❤♦s❡ ❝♦♥tr❛✲
❞✐❝t✐♦♥s ♦❜t❛✐♥❡❞ ❜❡❢♦r❡❀ t❤❡② s❡❡♠ t♦ ❜❡ ❛♥ ❛rt✐❢❛❝t ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ❧♦❣✐❝✱
♥♦t ✐♥ t✉♥❡ ✇✐t❤ Pr✐❡st✬s r❛③♦r✳
❚❤❡ ♣r❡✈✐♦✉s ❛r❣✉♠❡♥t ❛❣❛✐♥st t❤❡ r❛r✐t② ♦❢ ❞✐❛❧❡t❤❡✐❛s ✐s ❜❛s❡❞ ✐♥ t❤❡
tr❡❛t♠❡♥t ❣✐✈❡♥ ❜② ▲P t♦ ❣❧✉ts✳ ■t st✐❧❧ ♦♣❡♥s t❤❡ ❞♦♦r t♦ ❛ ✇✐❞❡r ♣r♦❜❧❡♠ ♦♥
t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❣❧✉ts ❛♥❞ ❞✐❛❧❡t❤❡✐❛s✳ ❈♦♥s✐❞❡r✐♥❣ ❛❣❛✐♥ t❤❡ s✐♠♣❧❡ ▲✐❛r
♣❛r❛❞♦①✱ L ✐s ❛ ❣❧✉t ❛♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ Pr✐❡st✱ ❛ ❞✐❛❧❡t❤❡✐❛✱ ❣✐✈❡♥ t❤❛t T ✭_L^✮ ∧
T ✭_¬L^✮✳ ❍♦✇❡✈❡r✱ ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ t❤❡ ❧♦❣✐❝ ▲P✱ t♦ ❜❡ ❛ ❣❧✉t ✐s s✐♠♣❧②
t♦ ❤❛✈❡ ❛ ❞❡✜♥✐t❡ tr✉t❤ ✈❛❧✉❡✱ ✇❤✐❝❤ ✐s ❜♦t❤ T ❛♥❞ F ✳ ❚❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ❝❛❧❧
❢♦r ♥❡❣❛t✐♦♥ ❤❡r❡✳ ❚❤❡ t②♣✐❝❛❧ ✈❛❧✉❛t✐♦♥ ✐♥ ▲P ❣♦❡s ❢r♦♠ ♣r♦♣♦s✐t✐♦♥s t♦ t❤❡
s❡ts ♦❢ tr✉t❤ ✈❛❧✉❡s {{T }, {F }, {T, F }}✳ ■♥ t❤✐s s❡♥s❡✱ ❛ s❡♥t❡♥❝❡ ♠❛② ❜❡ ❛ ❣❧✉t
s✐♠♣❧✐❝✐t❡r ✿ ✐t r❡❝❡✐✈❡❞ t❤❡ t❤✐r❞ ✈❛❧✉❡✳
❲❤❛t ✐s s♦ ❜❛❞ ❛❜♦✉t t❤❛t ❛❝❝♦✉♥t ✐s t❤❛t t❤❡ ✐❞❡❛ ♦❢ ❛ s❡♥t❡♥❝❡ r❡❝❡✐✈✐♥❣ ❛
❣❧✉t ❛s ✐ts tr✉t❤ ✈❛❧✉❡ ✐s ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ ♥❡❣❛t✐♦♥✳ ❊✈❡♥ ✐❢ ❛ ❧❛♥❣✉❛❣❡ ❤❛❞ ♥♦
♥❡❣❛t✐♦♥ s✐❣♥✱ s❡♥t❡♥❝❡s ✐♥ t❤❛t ✈❡r② ❧❛♥❣✉❛❣❡ ✇♦✉❧❞ st✐❧❧ ❜❡ ❛❜❧❡ t♦ ❜❡ ❣❧✉ts✳
❚❤❛t ✐s✱ ✐♥ ❛ ♥❡❣❛t✐♦♥✲❢r❡❡ ❢r❛❣♠❡♥t ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ▲P ✇❡ ✇♦✉❧❞ ❜❡ ❛❜❧❡
t♦ ♣r♦❞✉❝❡ ❣❧✉ts t❤❛t ♥❡❡❞ ♥♦t ❜❡ ❞✐❛❧❡t❤❡✐❛s✳ ❆❝❝❡♣t✐♥❣ ❣❧✉ts ✐s ❞✐st✐♥❝t ❢r♦♠
❛❝❝❡♣t✐♥❣ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s✳ ●❧✉ts ✇♦✉❧❞ st✐❧❧ ♠✉❧t✐♣❧② t❤❡♠s❡❧✈❡s✱ ❜✉t t❤❡r❡
✇♦✉❧❞ ❜❡ ♥♦ ❝♦♥tr❛❞✐❝t✐♦♥ ✐♥ t❤❡ s②♥t❛❝t✐❝❛❧ s❡♥s❡✱ ❛s ❞❡✜♥❡❞ ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣
♦❢ t❤❡ ♣❛♣❡r✳ ❚❤❛t ♣♦✐♥ts t♦ t❤❡ ❢❛❝t t❤❛t t❤❡r❡ ✐s s♦♠❡t❤✐♥❣ ♠✐ss✐♥❣ ✐♥ t❤❡
❞✐❛❧❡t❤❡✐sts✬ st♦r②✱ t❤❡ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ ❝❡♥tr❛❧ ❝♦♥❝❡♣ts ✐s ♥♦t ❛s ♦❜✈✐♦✉s ❛s
✐t ✐s t❛❦❡♥ t♦ ❜❡✳
❍♦✇❡✈❡r✱ t❤❡ ❛❜♦✈❡ ❛r❣✉♠❡♥t ❝♦✉❧❞ ❣♦ ♦♥ ❡✈❡♥ ✐♥ ❛ ❧❛♥❣✉❛❣❡ ❤❛✈✐♥❣ ❛
♥❡❣❛t✐♦♥ s✐❣♥✳ ❆s t❤❡ ❡①❛♠♣❧❡s ❛❜♦✈❡ ✐♥✈♦❧✈✐♥❣ ❝♦♥❥✉♥❝t✐♦♥ ❛♥❞ ❞✐s❥✉♥❝t✐♦♥
s❤♦✇✱ ♥♦t✐❝❡✱ t❤❛t ✈❡r② ❢❡✇ ♦❢ t❤♦s❡ s❡♥t❡♥❝❡s r❡♣r❡s❡♥t✐♥❣ ❣❧✉ts ❤❛✈❡ t❤❡ ❢♦r♠
♦❢ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❡ ♣r♦❜❧❡♠ ❝♦♠❡s ❛❣❛✐♥ ❢r♦♠ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♠❛❞❡ ❜②
t❤❡ ❞✐❛❧❡t❤❡✐st ❜❡t✇❡❡♥ ❣❧✉ts ❛♥❞ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s✳ ●✐✈❡♥ ♦♥❡ ❣❧✉t✱ ❛❝❝♦r❞✐♥❣
t♦ ▲P✱ ♥♦t ❡✈❡r② ♠♦❧❡❝✉❧❛r ❢♦r♠✉❧❛ t❤❛t ✐s ❛ ❣❧✉t ✇✐❧❧ ❜❡ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✦ ❙♦✱
t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❣❧✉ts ❛♥❞ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❢❛✐❧s ✐♥ t❤❡s❡ ❝❛s❡s t♦♦✱ ❛t
❧❡❛st s♦ ❢❛r ❛s t❤❡ ♦✣❝✐❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥ ❛♥❞ ❞✐❛❧❡t❤❡✐❛s ❣♦❡s✳ ❖♥❡
♠❛② ❤❛✈❡ ❣❧✉ts ✇✐t❤♦✉t tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s✱ ❛t ❧❡❛st ❛❝❝♦r❞✐♥❣ t♦ ▲P✳ P❡r❤❛♣s
♦♥❡ ❝♦✉❧❞ ❛✈♦✐❞ t❤❡ ❝❤❛r❣❡ ♦❢ ✐♥❝♦❤❡r❡♥❝❡ ❤❡r❡ ❜② ❞✐st✐♥❣✉✐s❤✐♥❣ ❣❧✉ts ❢r♦♠
❞✐❛❧❡t❤❡✐❛s✱ ❜✉t t❤❡♥ ❛ ❞✐❛❧❡t❤❡✐❝❛❧❧② ❛❝❝❡♣t❛❜❧❡ ✐♥❞❡♣❡♥❞❡♥t ♠♦t✐✈❛t✐♦♥ ❢♦r
s✉❝❤ ❛ ❞✐st✐♥❝t✐♦♥ ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ ♣r♦✈✐❞❡❞✱ ❛♥❞ ♠✉❝❤ ♦❢ ✇❤❛t ❤❛s ❛❧r❡❛❞② ❜❡❡♥
s❛✐❞ ❜② ❞✐❛❧❡t❤❡✐sts ✇✐❧❧ ❤❛✈❡ t♦ ❜❡ r❡✈✐s❡❞✳ ❍♦✇❡✈❡r✱ ♥♦t❤✐♥❣ t❤❛t ❞✐❛❧❡t❤❡✐sts
❤❛✈❡ ✇r✐tt❡♥ s♦ ❢❛r ✐♥❞✐❝❛t❡s t❤❡② ❛r❡ ❝♦♥s✐❞❡r✐♥❣ t❤♦s❡ ❝♦♥❝❡♣ts ❛s s❡♣❛r❛❜❧❡✱
s♦✱ ♦♥❝❡ ❛❣❛✐♥✱ t❤❡ ❞♦❝tr✐♥❡ s❡❡♠s t♦ ❧❛❝❦ ✐♥ ❝♦❤❡r❡♥❝❡✱ ❣✐✈❡♥ t❤❛t ✉♥✇❛rr❛♥t❡❞
✐❞❡♥t✐✜❝❛t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♠❛❞❡✳
❚❤❡ ❞✐❛❧❡t❤❡✐st ♠❛② ♣r♦t❡st ✐♥ t❤✐s ♣♦✐♥t✿ t❤❡ ♣r♦♣❡r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❢❛❧s✐t②
r❡q✉✐r❡s ♥❡❣❛t✐♦♥ ❛♥❞ ❛ tr✉t❤ ♣r❡❞✐❝❛t❡✳ ❙♦✱ ❡✈❡♥ ✐❢ ❢r♦♠ t❤❡ ❧♦❣✐❝❛❧ ♣♦✐♥t ♦❢
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✶✵✶
✈✐❡✇ ✐t ❞♦❡s ♠❛❦❡ s❡♥s❡ t♦ s❡♣❛r❛t❡ ❣❧✉ts ❢r♦♠ ❞✐❛❧❡t❤❡✐❛s✱ ❢r♦♠ ❛♥ ✐♥t✉✐t✐✈❡
♣♦✐♥t ♦❢ ✈✐❡✇ ✐t ❞♦❡s ♥♦t✳ ❚❤❡r❡ ✐s ❛♥ ❡①♣❧❛♥❛t♦r② r♦❧❡ t♦ ❜❡ ♣❧❛②❡❞ ❜② ♥❡❣❛t✐♦♥
✐♥ ❞❡✜♥✐♥❣ ❢❛❧s✐t②✳
❚♦ ❛♣♣r❡❝✐❛t❡ t❤❡ ♣♦✐♥t ♦❢ s✉❝❤ ❛ r❡♠❛r❦✱ ❧❡t ✉s ❞✐st✐♥❣✉✐s❤ t✇♦ ❛❝❝♦✉♥ts ♦❢
❣❧✉ts✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ✐♥❢♦r♠❛❧ ✐♥t✉✐t✐✈❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts✱ ❣❧✉ts ❛r❡ ❞❡♣❡♥❞❡♥t
♦❢ ♥❡❣❛t✐♦♥✱ t❤❡② ❛r❡✱ ✐♥ ❛ s❡♥s❡✱ ✏❞❡t❛❝❤❛❜❧❡✑ ✐♥ t✇♦ ❝♦♠♣♦♥❡♥ts✿ t❤❡ tr✉❡ ❛♥❞
t❤❡ ❢❛❧s❡✳ ■♥ t❤✐s ❝❛s❡✱ ❢♦r ❛ s❡♥t❡♥❝❡
T ✭_α^✮ ∧ T ✭_¬α^✮✳
α
t♦ ❜❡ ❛ ❣❧✉t ✐t ♠✉st ❜❡ t❤❡ ❝❛s❡ t❤❛t
❚❤❡ s✐♠♣❧❡ ▲✐❛r✱ ✐t ✐s s❛✐❞✱ ✐s ♦♥❡ s✉❝❤ ❣❧✉t✳ ❆❧t❡r♥❛t✐✈❡❧②✱
❛❝❝♦r❞✐♥❣ t♦ t❤❡ s❡❝♦♥❞ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts✱ t❤❡ ♦♥❡ ✇❡ ✜♥❞ ✐♥ ▲P✱ t❤❡② ❛r❡ ✏✉♥✲
❞❡t❛❝❤❛❜❧❡✑✱ ❛ttr✐❜✉t❡❞ ❛t ♦♥❝❡ ❛s ❛ s✐♥❣❧❡ ❡♥t✐t②✱ ❛s ✐t ✇❡r❡✱ t♦ ❛ ♣r♦♣♦s✐t✐♦♥✳
α t♦ ❜❡ ❛ ❣❧✉t
v(α) = {T, F }✳
■♥ t❤✐s ❝❛s❡✱ ❢♦r ❛ s❡♥t❡♥❝❡
♦♥❝❡✳ ❙♦✱ ✐❢
α
✐s ❛ ❣❧✉t✱
✐s t♦ ❜❡ ✈❛❧✉❡❞ ❜♦t❤ tr✉t❤ ✈❛❧✉❡s ❛t
■t ✐s t❤❡ s❡❝♦♥❞✱ ✉♥❞❡t❛❝❤❛❜❧❡ ❛❝❝♦✉♥t t❤❛t ❧❡❛❞s t♦ t❤❡ s❡♣❛r❛t✐♦♥ ♦❢ ❣❧✉ts
❢r♦♠ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛♥❞ t❤❡ ❧✐❦❡✳ ■t ✐s ❛❧s♦ t❤❡ s❡❝♦♥❞ t❤❛t ❧❡❛❞s ✉s ❞✐r❡❝t❧②
t♦ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❣❧✉ts ✭✇❤✐❝❤✱ ❛s ✇❡ ♠❡♥t✐♦♥❡❞✱ ♥❡❡❞s ♥♦t ❜❡ ❛❧s♦ ❛ ♠✉❧✲
t✐♣❧✐❝❛t✐♦♥ ♦❢ ❞✐❛❧❡t❤❡✐❛s✮✳ ❙♦✱ ✐❢ t❤❡ ❞✐❛❧❡t❤❡✐st ❝♦♠♣❧❛✐♥s t❤❛t t❤❡ ❞❡t❛❝❤❛❜❧❡✱
✐♥❢♦r♠❛❧ s❡♥s❡ ♦❢ ❣❧✉t ✐s t♦ ❜❡ ♣r❡❢❡rr❡❞✱ ✇❤✐❧❡ t❤❡ ✉♥❞❡t❛❝❤❛❜❧❡ s❡♥s❡ ✐s t♦ ❜❡
❛❜❛♥❞♦♥❡❞✱ t❤❡ ♣r♦❜❧❡♠ ❛r✐s❡s t❤❛t t❤❡ ▲♦❣✐❝ ♦❢ P❛r❛❞♦① ❞♦❡s ♥♦t r❡♣r❡s❡♥t t❤❡
✐♥t✉✐t✐♦♥s ❜❡❤✐♥❞ ❣❧✉ts✳ ❙♦✱ t❤❡r❡ ✐s ❛ ❧❛❝❦ ♦❢ ❤❛r♠♦♥② ✐♥ t❤❡ ❛❝❝♦✉♥t✱ t♦ s❛② t❤❡
❧❡❛st✱ ❛♥❞ ✇❡ ♠❛② r❡❛❧❧② ❣♦ ♦♥ ❛♥❞ s❛② t❤❛t t❤❡ ❞❡t❛❝❤❛❜❧❡ ❛♥❞ ✉♥❞❡t❛❝❤❛❜❧❡
❛❝❝♦✉♥ts ♦❢ ❣❧✉ts ❛r❡ ❞✐✛❡r❡♥t✳
❲❤❛t ✐s r❡❛❧❧② tr♦✉❜❧✐♥❣ ❛❜♦✉t t❤❛t ❞✐st✐♥❝t✐♦♥ ✐s t❤❛t ♦♥❡ ♦❢ t❤❡ ❞❡s✐❞❡r❛t❛
❢♦r ❛ ❞✐❛❧❡t❤❡✐st t❤❡♦r② ♦❢ ❧❛♥❣✉❛❣❡ ❛♥❞ tr✉t❤ ✐s t❤❛t ✐t s♦♠❡❤♦✇ ✇♦r❦s ❛s ❛
♠♦❞❡❧ ♦❢ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡ ❬✾✱ ♣♣✳✼✸✕✼✹❪✱ ❣❧✉ts ❛♥❞ ❞✐❛❧❡t❤❡✐❛s ✐♥❝❧✉❞❡❞✳ ❚❤❡r❡
❛r❡ s♦♠❡ ❢❡❛t✉r❡s ♦❢ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡ t❤❛t t❤❡ ❞✐❛❧❡t❤❡✐sts✬ ❧♦❣✐❝ s❤♦✉❧❞ ♠♦❞❡❧✳
●✐✈❡♥ t❤❡ ❛❜♦✈❡ ❝❧❛✐♠s t❤❛t t❤❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts ❡♥❝❛♣s✉❧❛t❡❞ ❜② ▲P ❢❛✐❧s
✐♥ ❝❛♣t✉r✐♥❣ s♦♠❡ ♦❢ t❤❡ ❢❡❛t✉r❡s ♦❢ ❣❧✉ts ✐♥ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡ ✭❛s t❤❡ ♥❡①t
❛r❣✉♠❡♥ts ❛❧s♦ ❛tt❡♠♣t t♦ ❡st❛❜❧✐s❤✮✱ t❤❡♥ t❤❡ ♠❛✐♥ r❡❛s♦♥ ❢♦r ❢✉r♥✐s❤✐♥❣ s✉❝❤
❛ ❧♦❣✐❝ ✈❛♥✐s❤✳ ❚❤❛t ✐s✱ ❢♦r♠❛❧ ❛♥❞ ✐♥❢♦r♠❛❧ ❛❝❝♦✉♥ts ♦❢ ❣❧✉ts ❛r❡ ♦✉t ♦❢ t✉♥❡✳
▲❡t ✉s ❝❤❡❝❦ ✇❤❛t t❤❡ ❞❡t❛❝❤❛❜❧❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts t❡❧❧s ✉s ❛❜♦✉t t❤❡ ❜❡❤❛✈✐♦r
♦❢ ❣❧✉ts✳
❆s ✇❡ s❤❛❧❧ s❡❡✱ t❤❛t ❛❝❝♦✉♥t ♦❢ ❣❧✉ts ❧❡❛❞s ✉s t♦ ❛♥♦t❤❡r ♣r♦❜❧❡♠
❢♦r ❞✐❛❧❡t❤❡✐s♠ ♦♥ ✇❤❛t ❝♦♥❝❡r♥s t❤❡ ♦♣❡r❛t✐♦♥ ♦❢ ❝♦♥♥❡❝t✐✈❡s ♦✈❡r ❣❧✉ts ❛♥❞✱
❛❣❛✐♥✱ ❝❛❧❧s ✐♥t♦ ❞♦✉❜t t❤❡ ❝♦❤❡r❡♥❝❡ ♦❢ t❤❡ ♣♦s✐t✐♦♥✳
▲❡t ✉s ❜❡❣✐♥ ✇✐t❤ t❤❡
♦♣❡r❛t✐♦♥ ♦❢ ♥❡❣❛t✐♦♥ ❛♥❞ ♥❡❣❛t✐♥❣ ❛ ❣❧✉t✳ ❲❤❛t ❞♦ ✇❡ ❡①♣❡❝t ✇❤❡♥ ✇❡ ♥❡❣❛t❡
❛ ❣❧✉t❄ ▼♦r❡ ❡❧❡♠❡♥t❛r✐❧② ♣✉t✿ ✇❤❡♥ ✇❡ ♥❡❣❛t❡ ❛ tr✉❡ ♣r♦♣♦s✐t✐♦♥✱ ✇❡ ✉s✉❛❧❧②
✉♥❞❡rst❛♥❞ ✐t ❛s ❤❛✈✐♥❣ t❤❡ ♦♣♣♦s✐t❡ tr✉t❤ ✈❛❧✉❡✱ ❢❛❧s❡✳ ❚❤❛t ❦✐♥❞ ♦❢ ✐♥t✉✐t✐♦♥
t❤❡ ❞✐❛❧❡t❤❡✐st ✇✐s❤❡s t♦ ♠❛✐♥t❛✐♥✱ ❛s ✇❤❡♥ Pr✐❡st ❬✾✱ ♣✳✻✹❪ s❛②s t❤❛t ✏❬♥❪❡❣❛t✐♦♥
✐s t❤❛t s❡♥t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ t✉r♥s ❛ tr✉❡ s❡♥t❡♥❝❡ ✐♥t♦ ❛ ❢❛❧s❡ ♦♥❡✱ ❛♥❞
✈✐❝❡ ✈❡rs❛✑ ✭s❡❡ ❛❧s♦ ❬✶✵✱ ❝❤❛♣✳✹❪✮✳
❍♦✇❡✈❡r✱ t❤✐s ✐♥t✉✐t✐♦♥ ✐s ♥♦t ❛♣♣❧✐❡❞ t♦
t❤❡ ♥❡❣❛t✐♦♥ ♦❢ ❛ ❣❧✉t✿ ✇❤❡♥ ✇❡ s❛② t❤❛t ❛ s❡♥t❡♥❝❡ ✐s ♥♦t ❛ ❣❧✉t✱ ✇❤❛t ❞♦ ✇❡
❡①♣❡❝t❄ ❚❤❛t ✐t ✐s ♥♦t ❜♦t❤ tr✉❡ ❛♥❞ ❢❛❧s❡✦ ❚❤❡ ❧♦❣✐❝ ▲P ♣r❡s❡♥ts ✉s ✇✐t❤ ❛
✶✵✷
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
❞✐✛❡r❡♥t ♣✐❝t✉r❡✱ ❛s ✇❡ ❛❧r❡❛❞② ♠❡♥t✐♦♥❡❞✳ ■❢ ✇❡ s❛② t❤❛t ❛ ❣❧✉tt② s❡♥t❡♥❝❡ ✐s
♥♦t t❤❡ ❝❛s❡✱ ✇❤❛t ✇❡ s❛✐❞ ✐s ✐ts❡❧❢ ❛ ❣❧✉t✳ ❚❤❡r❡ ❛r❡ t✇♦ ❝♦♥✢✐❝t✐♥❣ ✐♥t✉✐t✐♦♥s
r✉♥♥✐♥❣ ❜❡❤✐♥❞ t❤❡ s❝❡♥❡s ❤❡r❡✳
❚❤❡ ✉♥❞❡t❛❝❤❛❜❧❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts s❡❡♠s r❡s♣♦♥s✐❜❧❡ ❢♦r t❤❡ tr✉t❤ t❛❜❧❡ ♦❢
▲P✿ ✇❤❡♥ ✇❡ ♥❡❣❛t❡ ❛ s❡♥t❡♥❝❡ ✇❤♦s❡ ✈❛❧✉❡ ✐s {T, F }✱ t❤❡ ✐❞❡❛ s❡❡♠s t♦ ❜❡ t❤❛t
♥❡❣❛t✐♦♥ ♦♣❡r❛t❡s ♦✈❡r ❡❛❝❤ ♦❢ t❤❡ ✈❛❧✉❡s s❡♣❛r❛t❡❧②✱ ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ✈❛❧✉❡ ✐s
{F, T }✱ t❤❛t ✐s✱ t❤❡ s❛♠❡ s❡t ❛s ❜❡❢♦r❡✳ ❙♦✱ ♥❡❣❛t❡ ❛ ❣❧✉t ❛♥❞ ②♦✉ ❤❛✈❡ ❛ ❣❧✉t✳
❇✉t✱ ❛s ✇❡ ♠❡♥t✐♦♥❡❞ ❜❡❢♦r❡✱ t❤❡ ✐♥t✉✐t✐♦♥ ♦✈❡r ❣❧✉ts ❛♥❞ ♥❡❣❛t✐♦♥ s❡❡♠s t♦
♣♦✐♥t s♦♠❡✇❤❡r❡ ❡❧s❡✳ ▲❡t ✉s ❣♦ ✐♥t♦ ❛ ❢❡✇ ♠♦r❡ ❞❡t❛✐❧s✳
❑❡❡♣✐♥❣ ✇✐t❤ t❤❡ ✐♥t✉✐t✐✈❡ ❡①♣❧❛♥❛t✐♦♥ ♦❢ ❛ ❣❧✉t ✐♥ t❡r♠s ♦❢ ♥❡❣❛t✐♦♥ ❛♥❞
tr✉t❤✱ ❧❡t ✉s s❡❡ ❤♦✇ ♦✉r ✐♥t✉✐t✐♦♥s ❛❜♦✉t ♥❡❣❛t✐♦♥ s❤♦✉❧❞ ❜❡ ✈✐♥❞✐❝❛t❡❞✱ ❛❣❛✐♥st
✇❤❛t t❤❡ ❞✐❛❧❡t❤❡✐st ❤❡rs❡❧❢ s❛②s ✭❛♥❞ ❛❣❛✐♥st t❤❡ ✉♥❞❡t❛❝❤❛❜❧❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts
✐♥❝♦r♣♦r❛t❡❞ ✐♥ ▲P✮✳ ❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ ❣❧✉tt② s❡♥t❡♥❝❡ T ✭_L^✮ ∧ T ✭_¬L^✮ ♦❢
t❤❡ s✐♠♣❧❡ ▲✐❛r✳ ■❢ t❤✐s ✐s ❛ ❞✐❛❧❡t❤❡✐❛✱ ❜② ❞❡✜♥✐t✐♦♥ t❤✐s ✐s ❛ tr✉❡ st❛t❡♠❡♥t✱ ❛
❝♦♥❥✉♥❝t✐♦♥ ♦❢ t✇♦ tr✉❡ st❛t❡♠❡♥ts ✭❧❡t ✉s ❛ss✉♠❡ ✐t ❢♦r t❤❡ ♠♦♠❡♥t✱ ❜✉t s❡❡
❜❡❧♦✇✮✳ ❚❤✐s ♠❛② ❜❡ s❡❡♥ ✇❤❡♥ ✇❡ ♥♦t✐❝❡ t❤❛t t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦
T ✭_L ∧ ¬L^✮✳ ❚❤✐s ✐s ❛ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥✳ ◆❡❣❛t✐♥❣ ✐t✱ ✇❡ ❤❛✈❡ ¬T ✭_L ∧ ¬L^✮✳
■❢ t❤❡ ♦r✐❣✐♥❛❧ s❡♥t❡♥❝❡ ✇❛s s✐♠♣❧② tr✉❡ ✖ ❛s ✇❡ ❛r❡ ❛ss✉♠✐♥❣ ❢♦r t❤❡ s❛❦❡ ♦❢
❛r❣✉♠❡♥t ✖ ✐ts ♥❡❣❛t✐♦♥ ✐s s✐♠♣❧② ❢❛❧s❡ ✭t❤❡ ❞✐❛❧❡t❤❡✐st ✇♦✉❧❞ ❝♦♥❝❡❞❡ ✐t✱ s❡❡
❬✶✵✱ ❝❤❛♣✳✹❪✮✳ ❇② t❤❡ tr✉t❤ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ ❝♦♥❥✉♥❝t✐♦♥✱ t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ✐♥
❝❛s❡ ✐s ❢❛❧s❡ ✇❤❡♥ ♦♥❡ ❛♠♦♥❣ T ✭_L^✮ ❛♥❞ T ✭_¬L^✮ ✐s ❢❛❧s❡✱ t❤❛t ✐s✱ ✇❤❡♥ L ✐s ♥♦t
tr✉❡ ♦r ♥♦t ❢❛❧s❡✳ ❙♦✱ ♦♥❡ ♦❢ t❤❡ tr✉t❤ ✈❛❧✉❡s ♠✉st ❢❛✐❧ ❢♦r L✳ ❚❤❛t ✐s✱ ♥❡❣❛t✐♥❣
❛ ❣❧✉t ♠❛② ♥♦t ❡♥❞ ✉♣ ✐♥ ❛♥♦t❤❡r ❣❧✉t✳
❚❤❡ ✐ss✉❡ s❡❡♠s t♦ ❜❡ ❝❧❡❛r❡r ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ ❞✐❛❧♦❣✉❡✿
❲❤❛t ■ ❛♠ s❛②✐♥❣ ♥♦✇ ✐s ❜♦t❤ tr✉❡ ❛♥❞ ❢❛❧s❡✦
◆♦✱ ♥♦t ❜♦t❤✳
■♥ t❤✐s ❝❛s❡✱ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♥❡❣❛t✐♦♥ ♦✈❡r ❛ ❣❧✉t ❣✐✈❡s ✉s s♦♠❡t❤✐♥❣ t❤❛t
❝❧❡❛r❧② ✐s ♥♦t ❛ ❣❧✉t✳ ❚❤❡ ♣r♦❜❧❡♠ s❡❡♠s t♦ ❝♦♠❡ ❢r♦♠ ❛♥ ♦s❝✐❧❧❛t✐♦♥ ❜❡t✇❡❡♥
t❤❡ t✇♦ ❞✐st✐♥❝t tr❡❛t♠❡♥ts ♦❢ ❣❧✉ts✱ t❤❡ ❞❡t❛❝❤❛❜❧❡ ❛♥❞ t❤❡ ✉♥❞❡t❛❝❤❛❜❧❡✱ ♦❢
❝♦✉rs❡✳ ❚❤❡② ♣♦✐♥t t♦ ❞✐st✐♥❝t ❞✐r❡❝t✐♦♥s✳ ■♥ t❤❡ ❞❡t❛❝❤❛❜❧❡ ❝❛s❡✱ ✇❡ ♦♣❡r❛t❡
✇✐t❤ ♦t❤❡r ❝♦♥♥❡❝t✐✈❡s✱ ♠❛✐♥❧② ❝♦♥❥✉♥❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐♦♥✱ ❛♥❞ t❤❡✐r ✇♦r❦✐♥❣s
s❡❡♠ t♦ ♣r❡❝❧✉❞❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❝♦♥♥❡❝t✐✈❡s ✐♥ ▲P ✇❤❡♥ t❤❡② ❞❡❛❧ ✇✐t❤
❣❧✉tt② s❡♥t❡♥❝❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ✇❡ ❛❝❝❡♣t t❤❛t t❤❡ ♥❡❣❛t✐♦♥ ♦❢ ❛ ❣❧✉t ❧❡❛❞s t♦
❢❛❧s✐t②✱ ❛s t❤❡ ❛❜♦✈❡ ❞✐s❝✉ss✐♦♥ s✉❣❣❡sts✱ t❤❡ ❞✐❛❧❡t❤❡✐st ❧♦s❡s t❤❡ ♣❛r❛❝♦♥s✐st❡♥t
❝❤❛r❛❝t❡r ♦❢ t❤❡ ♥❡❣❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❡ss❡♥t✐❛❧ t♦ ❝♦♥tr♦❧ ❡①♣❧♦s✐♦♥ ✐♥ t❤❡ ❢❛❝❡
♦❢ ❝♦♥tr❛❞✐❝t✐♦♥s✳ ❍♦✇❡✈❡r✱ t❤❛t s❡❡♠s t♦ ❜❡ ❝❧♦s❡r t♦ t❤❡ ❢❛❝ts ✭r❡❝❛❧❧✱ ♦♥❝❡
❛❣❛✐♥✱ t❤❛t ♦♥❡ ♦❢ Pr✐❡st✬s ❣♦❛❧ ✐s t♦ ❦❡❡♣ ❝❧♦s❡ t♦ t❤❡ ❞❛t❛ ❢✉r♥✐s❤❡❞ ❜② ♥❛t✉r❛❧
❧❛♥❣✉❛❣❡s❀ ✇❡ t❤✐♥❦ t❤❡ ❛❜♦✈❡ ❞✐❛❧♦❣✉❡ ❝❧♦s❡ ❡♥♦✉❣❤ t♦ ✇❤❛t ✐s ❡①♣❡❝t❡❞ ❢r♦♠
♠♦st ♣❡♦♣❧❡ t❤❛t ❤❛✈❡ ♥♦t r❡❛❞ ❛❜♦✉t ❞✐❛❧❡t❤❡✐s♠✮✳
❉✐❛❧❡t❤❡✐st✿
▼♦♥❛❧❡t❤❡✐st✿
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✶✵✸
■❢ ♦♥❡ ✐s ♥♦t ❝♦♥✈✐♥❝❡❞ ♦❢ t❤❡ ❞♦✉❜❧❡ st❛♥❞❛r❞ t❤❛t ✐s ♦♣❡r❛t✐♥❣ ❤❡r❡✱ ❝♦♥✲
s✐❞❡r ❛❣❛✐♥ t❤❡ s✐♠♣❧❡ ▲✐❛r✳ ❆❝❝♦r❞✐♥❣ t♦ Pr✐❡st✱ ✐ts ❞❡✜♥✐♥❣ s❡♥t❡♥❝❡
L
✐s ❛
❣❧✉t✳ ■♥ ▲P✱ ♦♥❡ ♠❛② ❛❧✇❛②s ♥❡❣❛t❡ ❛ ❣❧✉t✱ ❛♥❞ t❤❡ r❡s✉❧t ✐s ❛ ❣❧✉t✳ ❍♦✇❡✈❡r✱
r❡❝❛❧❧ ♦✉r ❞✐s❝✉ss✐♦♥ ✐♥ s❡❝t✐♦♥ ✷ ♦♥ t❤❡ ❡①t❡♥❞❡❞ ❛♥❞ t❤❡ s✐♠♣❧❡ ▲✐❛r✳
❚❤❡
♠❛✐♥ ❢❡❛t✉r❡ ♦❢ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ s✐♠♣❧❡ ▲✐❛r✱ ❛❝❝♦r❞✐♥❣ t♦ Pr✐❡st✱ ✐s t❤❛t
✇❡ ❝❛♥♥♦t ❧❡❣✐t✐♠❛t❡❧② ❤❛✈❡ ✐ts ♥❡❣❛t✐♦♥✦ ❲❤② ♥♦t❄ ❇❡❝❛✉s❡✱ ❜❡✐♥❣ ❛ ❝♦♥❥✉♥❝✲
t✐♦♥ ♦❢ t✇♦ tr✉❡ st❛t❡♠❡♥ts ✭❛t ❧❡❛st t❤❡ ❞✐❛❧❡t❤❡✐st ❛ss✉♠❡s t❤❛t✮✱ ✐ts ♥❡❣❛t✐♦♥
✇♦✉❧❞ ❜❡ ❢❛❧s❡✱ ✇❡ ♣r❡s✉♠❡✳
❙♦✱ t❤❡r❡ ✐s ❛ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ tr❡❛t♠❡♥t
♦❢ ❣❧✉ts ✐♥ t❤❡ ❢♦r♠❛❧ ❧♦❣✐❝ ▲P ❛♥❞ ✐♥ t❤❡ ✐♥❢♦r♠❛❧ ♣❛r❛❞✐❣♠❛t✐❝ ♣r♦♦❢ ❢♦r t❤❡
❡①✐st❡♥❝❡ ♦❢ ❣❧✉ts✱ t❤❡ s✐♠♣❧❡ ▲✐❛r✳
❚❤❡ ❞❡t❛❝❤❛❜❧❡ ❝❤❛r❛❝t❡r ♦❢ ❣❧✉ts ❣❡♥❡r❛t❡s ♦t❤❡r ❞✐s❝r❡♣❛♥❝✐❡s ✇✐t❤ ❝♦♥✲
❥✉♥❝t✐♦♥ ❛♥❞ ❞✐s❥✉♥❝t✐♦♥ ❛s ✇❡❧❧✳ ❈♦♥s✐❞❡r ❛❣❛✐♥ ♦✉r s✐♠♣❧❡ ▲✐❛r
❚❛❦✐♥❣ ❡①♣❧✐❝✐t❧② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❣❧✉t ❛s ❛ ❝♦♥❥✉♥❝t✐♦♥✱ ✭T ✭_L^✮
L ❛s ❛ ❣❧✉t✳
∧ T ✭_¬L^✮✮✱
✇❤❛t ✇❡ ❤❛✈❡ ✐s ❛ ❝♦♥❥✉♥❝t✐♦♥ ❜❡t✇❡❡♥ ❛ tr✉❡ ❛♥❞ ❛ ❢❛❧s❡ s❡♥t❡♥❝❡✳ ❆❝❝♦r❞✐♥❣
t♦ t❤❡ tr✉t❤ t❛❜❧❡s ❢♦r ❝♦♥❥✉♥❝t✐♦♥ ✐♥ ▲P ✭✇❤✐❝❤ ❤❛♣♣❡♥s t♦ ❜❡ t❤❡ s❛♠❡ ❛s
✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✐♥ t❤✐s ❝❛s❡✮✱ t❤❡ r❡s✉❧t s❤♦✉❧❞ ❜❡ ❢❛❧s❡✳ ❙♦✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡
❞❡t❛❝❤❛❜❧❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❣❧✉t✱ t❤❡ ❣❧✉t s❡♥t❡♥❝❡ ♦❢ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ s✐♠♣❧❡
▲✐❛r ✐s s✐♠♣❧② ❢❛❧s❡✱ ❛s t❤❡ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✐❛♥ ✇♦✉❧❞ ❤❛✈❡ ✐t✳ ❈♦♥s✐❞❡r ❛❣❛✐♥ ♦✉r
S ❢♦r ✏s♥♦✇ ✐s ✇❤✐t❡✑✳ ■❢ ✇❡ ❛♣♣❧② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❣❧✉t ❢♦r t❤❡
L ∧ S ✱ ✇❡ ❤❛✈❡ ✭T ✭_L^✮ ∧ T ✭_¬L^✮✮ ∧ T ✭_S^✮✱ ✇❤✐❝❤ ❛❣❛✐♥ ✐s ❛ ❢❛❧s❡
tr✉❡ s❡♥t❡♥❝❡
❝♦♥❥✉♥❝t✐♦♥
s❡♥t❡♥❝❡✱ ♥♦t ❛ ❣❧✉t✳ ❚❤❛t ✐s ❛ r❡s✉❧t ♦❢ t❤❡ ❢❛❧s✐t② ♦❢ t❤❡ ❣❧✉t s❡♥t❡♥❝❡✳ ❙♦✱
❝♦♥tr❛r✐❧② ❢r♦♠ ✇❤❛t t❤❡ tr✉t❤ t❛❜❧❡s ♦❢ ▲P s❛②✱ t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❛ ❣❧✉t ✇✐t❤
❛♥② tr✉❡ s❡♥t❡♥❝❡ ✐s ♥♦t ❛❣❛✐♥ ❛ ❣❧✉t✱ ❜✉t r❛t❤❡r ❛ ❢❛❧s✐t②✱ ✇❤❡♥ s❡❡♥ ❢r♦♠ t❤✐s
♣♦✐♥t ♦❢ ✈✐❡✇✳
❉✐s❥✉♥❝t✐♦♥ ♣r❡s❡♥ts s✐♠✐❧❛r ❞✐✣❝✉❧t✐❡s✳ ▲❡t ✉s t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ❛❣❛✐♥ ♦✉r
G ❢♦r ✏s♥♦✇ ✐s ❣r❡❡♥✑✳ ❚❤❡♥✱ ✏ L ∨ G✑✱ ❢♦❧❧♦✇✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s✱
❜❡❝♦♠❡s ✏✭T ✭_L^✮ ∧ T ✭_¬L^✮✮ ∨ F ✭_G^✮✮✑✳ ❉✐str✐❜✉t✐♥❣ ♦✈❡r ❞✐s❥✉♥❝t✐♦♥✱ ✇❡ ❣❡t
✏✭T ✭_L^✮ ∨ F ✭_G^✮✮ ∧ ✭T ✭_¬L^✮ ∨ F ✭_G^✮✮✑✳ ❚❤❡ r❡s✉❧t ✐s ❛❣❛✐♥ ❛ ❢❛❧s❡ s❡♥t❡♥❝❡✱
❢❛❧s❡ s❡♥t❡♥❝❡
♣r♦✈✐❞❡❞ ✇❡ ❢♦❧❧♦✇ t❤❡ tr✉t❤ t❛❜❧❡s ♦❢ ▲P ✇❤❡♥ ❞❡❛❧✐♥❣ ✇✐t❤ ❝❧❛ss✐❝❛❧ tr✉t❤
✈❛❧✉❡s✳ ❚❤❡ ❡①♣❧❛♥❛t✐♦♥ ✐s s✐♠♣❧❡✿ ❥✉st ♥♦t✐❝❡ t❤❛t t❤❡ s❡❝♦♥❞ ❝♦♥❥✉♥❝t ✐s ✐♥
❢❛❝t ❛ ❞✐s❥✉♥❝t✐♦♥ ♦❢ t✇♦ ❢❛❧s❡ s❡♥t❡♥❝❡s✱
❛♥❞
F ✭_G^✮✳
T ✭_¬L^✮ ✭✇❤✐❝❤ ✐s ❜② ❞❡✜♥✐t✐♦♥ F ✭_L^✮✮✱
❍♦✇❡✈❡r✱ ❛s ✇❡ ♠❡♥t✐♦♥❡❞ ❜❡❢♦r❡✱ t❤❡ ❞✐s❥✉♥❝t✐♦♥ ♦❢ ❛ ❣❧✉t ❛♥❞ ❛
❢❛❧s✐t② ✐s ❛ ❣❧✉t ❛❝❝♦r❞✐♥❣ t♦ ▲P✱ ❜✉t ♥♦t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ✇❤❡♥ t❤❡
❞❡t❛❝❤❛❜❧❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❣❧✉t ✐s ❡♠♣❧♦②❡❞✳
❚❤❡ ♣r♦❜❧❡♠ ❝♦♠❡s ❢r♦♠ ❛♥ ❛♠❜✐❣✉✐t② ❜❡t✇❡❡♥ t❤❡ ❞❡t❛❝❤❛❜❧❡ ❛♥❞ t❤❡ ✉♥✲
❞❡t❛❝❤❛❜❧❡ ✈❡rs✐♦♥s ♦❢ ❣❧✉ts✳ ■♥ ▲P t❤❡ ❣❧✉ts ❛r❡ tr❡❛t❡❞ ❛s ✉♥❞❡t❛❝❤❛❜❧❡✱ t❤❡②
❛r❡ ❛ttr✐❜✉t❡❞
❡♥ ❜❧♦❝
t♦ t❤❡ s❡♥t❡♥❝❡s✱ ✐♥❞❡♣❡♥❞❡♥t❧② ❡✈❡♥ ♦❢ ♥❡❣❛t✐♦♥✳ ■♥❢♦r✲
♠❛❧❧②✱ ❢♦❧❧♦✇✐♥❣ t❤❡ s✐♠♣❧❡ ▲✐❛r✱ ✇❤✐❝❤ ✐s t❤❡ ♠❛✐♥ ♠♦t✐✈❛t✐♦♥ ❢♦r ❞✐❛❧❡t❤❡✐s♠✱
❣❧✉ts ❛r❡ ❛ttr✐❜✉t❡❞ t♦ s❡♥t❡♥❝❡s t❤❛t ❝♦♠❡ ✐♥ ❝♦♥❥✉♥❝t✐♦♥s✱ ❛♥❞ tr❡❛t❡❞ ❛s ❞❡✲
t❛❝❤❛❜❧❡✳ ■♥ t❤❡ ✜rst ❝❛s❡✱
L
✐ts❡❧❢ r❡❝❡✐✈❡s t❤❡ ❣❧✉t tr✉t❤ ✈❛❧✉❡✱ ✐♥ t❤❡ s❡❝♦♥❞
❝❛s❡✱ ✇❡ ❛r❡ ♥♦ ❧♦♥❣❡r ❞❡❛❧✐♥❣ ❞✐r❡❝t❧② ✇✐t❤
L✱
❜✉t ✇✐t❤ ❛ ❝♦♠♣♦✉♥❞ s❡♥t❡♥❝❡
✶✵✹
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
t❤❛t ✐♥✈♦❧✈❡s L ❛♥❞ ✐ts tr✉t❤ ✈❛❧✉❡s✳
❙♦✱ ♣❡r❤❛♣s t❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t ❛ ❝♦❤❡r❡♥t ❛❝❝♦✉♥t ♦❢ t❤❡ ❜❡❤❛✈✐♦r ♦❢
❣❧✉ts st✐❧❧ ❝❤❛❧❧❡♥❣❡s t❤❡ ❞✐❛❧❡t❤❡✐st✳ ■♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ▲P ✐t ❜❡❤❛✈❡s ♦♥❡ ✇❛②✱
✐♥ t❤❡ ♠❡t❛❧❛♥❣✉❛❣❡✱ ✐t ❜❡❤❛✈❡s ❞✐✛❡r❡♥t❧②✳ ❍♦✇❡✈❡r✱ t❤❡ ❞✐❛❧❡t❤❡✐st ✇✐s❤❡s
t♦ ❛✈♦✐❞ s✉❝❤ ❛ ❞✐✈✐s✐♦♥ ❜❡t✇❡❡♥ ❧❡✈❡❧s ✐♥ ❧❛♥❣✉❛❣❡✱ ✇❤✐❝❤ s❤❡ ❞❡❡♠s ❛rt✐✜❝✐❛❧
✭❢♦r ✐♥st❛♥❝❡✱ s❡❡ ❬✾✱ ❝❤❛♣✳✶❪✮✳ ❲❤❛t t❤❡♥❄ ❍♦✇ ❛r❡ ✇❡ s✉♣♣♦s❡❞ t♦ ✉♥❞❡rst❛♥❞
❣❧✉ts❄ ❚❤♦s❡ ❛r❡ ❞✐✣❝✉❧t q✉❡st✐♦♥s✱ ❛♥❞ ❤♦✇❡✈❡r ✇❡ ❛♥s✇❡r t❤❡♠✱ t❤❡ ❜❡❤❛✈✐♦r
♦❢ ❣❧✉ts✱ ❛s ✇❡ ❤❛✈❡ s❡❡♥ s♦ ❢❛r✱ s❡❡♠s ❢❛r ❢r♦♠ t❤❡ ❞❡s✐r❡❞ ✐♥t✉✐t✐♦♥ ✉♥❞❡r❧②✐♥❣
❞✐❛❧❡t❤❡✐s♠✳
✹ ❚❤❡ ❢❛❧s✐t② ♦❢ ❛ ❣❧✉t
❆s ✇❡ ❤❛✈❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ❛ s❡♥t❡♥❝❡ ❡①♣r❡ss✐♥❣ ❛ ❣❧✉t✱
❥✉❞❣✐♥❣ ❢r♦♠ ✐ts ❞❡✜♥✐t✐♦♥✱ ✐s ❛❧✇❛②s ❢❛❧s❡✳ ❚❤❡ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✐❛♥ ❝♦✉❧❞✱ ♦❢
❝♦✉rs❡✱ s❛② t❤❛t s❤❡ ❛❧r❡❛❞② ❦♥❡✇ t❤❛t✳ ❚❤❡ ❞✐❛❧❡t❤❡✐st✱ ❛s Pr✐❡st✱ ❝♦✉❧❞ ❝❧❛✐♠
t❤❛t ❤❡ ❤❛❞ ❛♥t✐❝✐♣❛t❡❞ t❤❛t ♦❜❥❡❝t✐♦♥✳ ❆❝❝♦r❞✐♥❣ t♦ ❛♥ ♦❜❥❡❝t✐♦♥ t❤❛t Pr✐❡st
❬✾✱ ♣✳✾✾❪ ❞❡❛❧s ✇✐t❤✱ ❤❛✈✐♥❣ t❤❡ ❜❡❧✐❡❢ ✐♥ ❣❧✉tt② ❝♦♥tr❛❞✐❝t✐♦♥s ✇✐❧❧ ❢♦r❝❡ ✉s t♦
❜❡❧✐❡✈❡ s♦♠❡ ❢❛❧s❡ ♣r♦♣♦s✐t✐♦♥s✱ ❜❡❝❛✉s❡ ❣❧✉ts ❛r❡ ❢❛❧s❡ t♦♦✳ ❙♦✱ ❤♦✇ ❞❛♠❛❣✐♥❣
✐s t❤❡ ♠❛✐♥ ♦❜❥❡❝t✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥❄
❖✉r ❛❜♦✈❡ ♦❜❥❡❝t✐♦♥ ✐s ♠✉❝❤ ♠♦r❡ s❡r✐♦✉s t❤❛♥ t❤❡ ♦♥❡ ❛♥t✐❝✐♣❛t❡❞ ❜②
Pr✐❡st✳ ▲❡t ✉s ❝❤❡❝❦ ❤♦✇ Pr✐❡st ❞❡❛❧s ✇✐t❤ t❤❡ ✐ss✉❡✳ ❆❝❝♦r❞✐♥❣ t♦ Pr✐❡st✱ t♦
❜❡❧✐❡✈❡ ✐♥ s♦♠❡ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s ✐♥✈♦❧✈❡s ❜❡❧✐❡✈✐♥❣ t❤❛t t❤❡② ❛r❡ tr✉❡ ❛♥❞
t❤❛t t❤❡② ❛r❡ ❢❛❧s❡✳ ❚❤✐s ✐s ❡✈✐❞❡♥❝❡ t❤❛t Pr✐❡st tr❡❛ts ❣❧✉ts ❛s ❞❡t❛❝❤❛❜❧❡✳
❆ ♣❛rt ♦❢ t❤❡ ❣❧✉tt② s❡♥t❡♥❝❡ ✐s tr✉❡✳ ❆♥♦t❤❡r ♣❛rt ♦❢ t❤❡ ❣❧✉tt② s❡♥t❡♥❝❡ ✐s
❢❛❧s❡✳ ❨♦✉ ❜❡❧✐❡✈❡ ✐♥ ❡❛❝❤ ❝♦♥❥✉♥❝t ♦❢ ❛ ❝♦♥❥✉♥❝t✐♦♥ ✇❤❡♥ ②♦✉ ❜❡❧✐❡✈❡ ✐♥ t❤❡
✇❤♦❧❡ ❝♦♥❥✉♥❝t✐♦♥✳ ❙♦✱ ②♦✉ ❜❡❧✐❡✈❡ ✐♥ s♦♠❡ ❢❛❧s✐t✐❡s t♦♦✳ ❇✉t t❤❛t ✐s ♥♦t ❛
♣r♦❜❧❡♠✱ ❜❡❝❛✉s❡ ❜❡❧✐❡❢ ✐♥ ❛ ❣❧✉t ✐s ❜❡❧✐❡❢ ✐♥ ❛ tr✉❡ ♣r♦♣♦s✐t✐♦♥ ❛s ✇❡❧❧✳ ●✐✈❡♥
t❤❛t ❛❝❝♦r❞✐♥❣ t♦ Pr✐❡st tr✉t❤ ❞♦♠✐♥❛t❡s ♦✈❡r ❢❛❧s✐t②✱ ✇❤❛t❡✈❡r t❤❛t ♠❡❛♥s✱
t❤❡♥ t❤❡r❡ ✐s ♥♦ ♣r♦❜❧❡♠ ✐♥ ❤❛✈✐♥❣ s♦♠❡ ❢❛❧s❡ ❜❡❧✐❡❢s✱ ♣r♦✈✐❞❡❞ t❤❛t t❤❡② ❛r❡
❛❧s♦ tr✉❡✳✽
❖✉r ♦❜❥❡❝t✐♦♥ ❣♦❡s str❛✐❣❤t ❛❣❛✐♥st s✉❝❤ ❛♥ ❛♥s✇❡r✿ ♦✉r ❛r❣✉♠❡♥t ♣♦✐♥ts t♦
t❤❡ ❢❛❝t t❤❛t ✐♥ t❤❡ ❞❡t❛❝❤❛❜❧❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❣❧✉ts t❤❡ ❝♦♥❥✉♥❝t✐♦♥ T ✭_L^✮ ∧
T ✭_¬L^✮ s❤♦✉❧❞ ❜❡ ✉♥❞❡rst♦♦❞ ❛s ♣❧❛✐♥❧② ❢❛❧s❡✱ ❛s ❛♥② ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❛ tr✉t❤ ❛♥❞
❛ ❢❛❧s✐t② ✇♦✉❧❞ ❜❡✱ ♥♦ tr✉t❤ ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ tr✉t❤ ✈❛❧✉❡ ♦❢ t❤❡ ✇❤♦❧❡ ❝♦♥❥✉♥❝t✐♦♥✳
■t ✐s ❢❛❧s❡✱ ❛♥❞ ♦♥❧② ❢❛❧s❡✳ ❚❤❛t ✐s ❤♦✇ t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ❜❡❤❛✈❡s ✐♥ ▲P t♦♦✱
✽
❖❢ ❝♦✉rs❡✱ t❤❛t ❝❧❛✐♠ st✐❧❧ ❞♦❡s ♥♦t ❛❞❞r❡ss t❤❡ ✐ss✉❡ ♦❢ ✇❤❛t ❛r❡ ✇❡ t♦ ❞♦ ✇✐t❤ t❤❡
❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r✳ Pr✐❡st ❬✽✱ ♣✳✸✹❪ s✉❣❣❡sts t❤❛t ✇❡ s❤♦✉❧❞ ♥♦t ❜❡❧✐❡✈❡ ✐♥ ✉♥tr✉❡
s❡♥t❡♥❝❡s✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r ✐s ❜♦t❤ tr✉❡ ❛♥❞ ♥♦t✲tr✉❡ ✭✉♥tr✉❡✮✳
❚❤❛t s✐t✉❛t✐♦♥ ❝♦♥tr✐❜✉t❡s t♦ t❤❡ ❝❧❛✐♠ t❤❛t ❝♦❤❡r❡♥❝❡ ✐s ❧❛❝❦✐♥❣ ✐♥ t❤❡ ❞✐❛❧❡t❤❡✐st ♣✐❝t✉r❡✳
■♥t❡r❡st✐♥❣ ❛s ✐t ✐s✱ ❤♦✇❡✈❡r✱ ✇❡ s❤❛❧❧ ♥♦t ❞✐s❝✉ss t❤❡ ✐ss✉❡ ♦❢ ❜❡❧✐❡❢ ✐♥ ✉♥tr✉❡ ♣r♦♣♦s✐t✐♦♥s
❤❡r❡✳
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✶✵✺
✇❤❡♥ ❞❡❛❧✐♥❣ ♦♥❧② ✇✐t❤ ❝❧❛ss✐❝❛❧ tr✉t❤ ✈❛❧✉❡s✳ ❙♦✱ ❛❝❝♦r❞✐♥❣ t♦ ♦✉r r❡❛❞✐♥❣ ♦❢
t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ s✐♠♣❧❡ ▲✐❛r✱ t❤❛t ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ♣❛r❛❞♦①✱ t❛❦❡♥ ❛s ❛
❝♦♥❥✉♥❝t✐♦♥✱ ✐s s✐♠♣❧② ❢❛❧s❡✱ ❛s ❛ ❝❧❛ss✐❝❛❧ ❝♦♥tr❛❞✐❝t✐♦♥✳
P❡r❤❛♣s ✐t ❝♦✉❧❞ ❜❡ ❛r❣✉❡❞ t❤❛t t❤❡ ❛❜♦✈❡ ❛r❣✉♠❡♥t s❤♦✇s ♦♥❧② t❤❛t✱ ✉♥❞❡r
t❤❡ ❞❡t❛❝❤❛❜❧❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❣❧✉ts✱ t❤❡r❡ ✐s ♥♦ ♠❡r❡❧② tr✉❡ ♣r❡❞✐❝❛t✐♦♥ ♦❢
s♦♠❡t❤✐♥❣ ❛s ❜❡✐♥❣ ❛ ❣❧✉t✳ ❚❤❛t ✐s✱ t❤❡ ♦r✐❣✐♥❛❧ s❡♥t❡♥❝❡ T ✭_L^✮ ∧ T ✭_¬L^✮
✐s ♥❡✈❡r ♦♥❧② tr✉❡✱ ❜✉t ❛❧✇❛②s ❜♦t❤ tr✉❡ ❛♥❞ ❢❛❧s❡✱ s♦ t❤❛t t❤❡ ❛ttr✐❜✉t✐♦♥ ♦❢
❣❧✉t ❝❤❛r❛❝t❡r t♦ L ✐s ✐ts❡❧❢ ❛ ❣❧✉tt② s❡♥t❡♥❝❡✳ ❚❤❡r❡ ❛r❡ t✇♦ ♣r♦❜❧❡♠s ✇✐t❤
t❤❛t s✉❣❣❡st✐♦♥✳ ❚❤❡ ✜rst ♦♥❡ ✐s t❤❛t t❤❡r❡ ✇✐❧❧ ❜❡ ♥♦ tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉t
♦♥❧② tr✉❡ ❛♥❞ ❢❛❧s❡ ❝♦♥tr❛❞✐❝t✐♦♥s✳ ❚❤❛t ✐s ❝❧❡❛r❧② ❛❣❛✐♥st t❤❡ ❣❡♥❡r❛❧ t❡♥❡t
♦❢ ❞✐❛❧❡t❤❡✐s♠✱ ✇❤✐❝❤ ❝❧❡❛r❧② ✐s ❞❡✜♥❡❞ ❛s st❛t✐♥❣ t❤❛t s♦♠❡ ❝♦♥tr❛❞✐❝t✐♦♥s ❛r❡
tr✉❡✱ ♦r ❛t ❧❡❛st ✐t s❡❡♠s s♦✳ ❇✉t t❤❛t ✐s ❛ ♠✐♥♦r ♣r♦❜❧❡♠✱ ✇❤✐❝❤ ♠❛② ❜❡
s♦❧✈❡❞ ❜② ❛ ❝❤❛♥❣❡ ✐♥ t❡r♠✐♥♦❧♦❣②✳ ❚❤❡ s❡❝♦♥❞ ❞✐✣❝✉❧t② ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t
t❤❛t t❤❡ ♣r❡✈✐♦✉s s✉❣❣❡st✐♦♥ ✐s ♦❢ ♥♦ ❤❡❧♣ t♦ t❤❡ ❞✐❛❧❡t❤❡✐st✿ t❤❡ ✐❞❡❛ t❤❛t
❞✐❛❧❡t❤❡✐❛s ❛r❡ tr✉❡ ❛♥❞ ❢❛❧s❡✱ ❛s s✉❣❣❡st❡❞✱ ♦♣❡r❛t❡s ♦♥ ❛♥ ❡q✉✐✈♦❝❛t✐♦♥ ♦✈❡r
T ✭_¬L^✮ ❛♥❞ F ✭_L^✮✳ ❲❤❡♥ t❤❡ s❡❝♦♥❞ ❝♦♥❥✉♥❝t ✐♥ T ✭_L^✮ ∧ T ✭_¬L^✮ ✐s s❡❡♥ ❛s
T ✭_¬L^✮✱ ✇❡ s❡❡♠ t♦ ❤❛✈❡ ❛ ❝♦♥❥✉♥❝t✐♦♥ ❜❡t✇❡❡♥ t✇♦ tr✉❡ s❡♥t❡♥❝❡s✱ ❛♥❞ ✇❡
❤❛✈❡ tr✉t❤ ❛s ❛ r❡s✉❧t✳ ❲❤❡♥ t❤❡ s❡❝♦♥❞ ❝♦♥❥✉♥❝t ✐s s❡❡♥ ❛s F ✭_L^✮✱ ✇❡ ❤❛✈❡ ❛
❝♦♥❥✉♥❝t✐♦♥ ❜❡t✇❡❡♥ tr✉t❤ ❛♥❞ ❢❛❧s❡❤♦♦❞✱ r❡s✉❧t✐♥❣ ✐♥ ❛ ❢❛❧s❡ s❡♥t❡♥❝❡✳ ❍♦✇❡✈❡r✱
t❤❛t ❛♠❜✐❣✉✐t② ✐s ❡❛s✐❧② s♦❧✈❡❞ ✇❤❡♥ ✇❡ r❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❢❛❧s❡❤♦♦❞✿ ✇❡
r❡❛❧❧② ♦♥❧② ❤❛✈❡ ❛ ❝♦♥❥✉♥❝t✐♦♥ ❜❡t✇❡❡♥ ❛ tr✉❡ ❛♥❞ ❛ ❢❛❧s❡ s❡♥t❡♥❝❡✱ ❛♥❞ t❤❡
r❡s✉❧t ♦❢ t❤❡ ❝♦♥❥✉♥❝t✐♦♥✱ t❛❦✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❢❛❧s❡❤♦♦❞ s❡r✐♦✉s❧② t❤❡♥ ❞♦❡s
♥♦t ♦s❝✐❧❧❛t❡ ❜❡t✇❡❡♥ tr✉t❤ ❛♥❞ ❢❛❧s❡❤♦♦❞✱ ✐t ✐s ♦♥❧② ❢❛❧s❡✳✾
◆♦t✐❝❡✱ ❢✉rt❤❡r♠♦r❡✱ t❤❛t Pr✐❡st ❤✐♠s❡❧❢ ✐s ❛♠❜✐❣✉♦✉s ❛s t♦ ❤♦✇ t♦ tr❡❛t
❣❧✉tt② s❡♥t❡♥❝❡s✳ ❙♦♠❡t✐♠❡s ❤❡ ❞❡❛❧s ✇✐t❤ t❤❡♠ ❛s ✐❢ t❤❡ t✇♦ tr✉t❤ ✈❛❧✉❡s ✇❡r❡
s❡♣❛r❛❜❧❡✱ s♦♠❡t✐♠❡s✱ ❛s ✐❢ t❤❡② ✇❡r❡ ❜♦t❤ t♦❣❡t❤❡r ❛ttr✐❜✉t❡❞ t♦ ❛ s❡♥t❡♥❝❡✱ ❛s
❛ ✉♥✐t✳ ❚❤❡ ✜rst tr❡❛t♠❡♥t ❛❝❝♦✉♥ts ❢♦r t❤❡ ❝❧❛✐♠ t❤❛t ✐t ✐s ❧❡❣✐t✐♠❛t❡ t♦ ✐♥❢❡r
t❤❛t L ✐s tr✉❡ ❢r♦♠ T ✭_L^✮ ∧ T ✭_¬L^✮✱ ❛♥❞ ✐t ✐s ❛❧s♦ ❧❡❣✐t✐♠❛t❡ t♦ ✐♥❢❡r t❤❛t L
✐s ❢❛❧s❡ ❢r♦♠ T ✭_L ^✮ ∧ T ✭_¬L^✮✳ ❚❛❦✐♥❣ t❤❛t ❢♦r ❣r❛♥t❡❞✱ ❛♥❞ ❝♦♥tr❛r② t♦ ✇❤❛t
✇❡ ❤❛✈❡ ❛r❣✉❡❞ ❜❡❢♦r❡✱ ❧❡t ✉s ❝♦♥❝❡❞❡ t❤❛t t❤❡ s❡♥t❡♥❝❡ T ✭_L^✮ ∧ T ✭_¬L^✮ ✐s
❜♦t❤ tr✉❡ ❛♥❞ ❢❛❧s❡✱ ❛♥❞ t❤❛t ✐t ❧✐❡s ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ t❤❡ s✐♠♣❧❡
▲✐❛r✳ ❚❤❡ ❞✐❛❧❡t❤❡✐st st♦♣s t❤❡r❡ ❛♥❞ ❝❧❛✐♠s t❤❛t s♦♠❡ s❡♥t❡♥❝❡s ❛r❡ ❜♦t❤ tr✉❡
❛♥❞ ❢❛❧s❡✳ ❖✉r ❝❧❛✐♠ ✐s t❤❛t✱ ✉♥❞❡r t❤❡ ❞❡t❛❝❤❛❜❧❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❣❧✉ts✱ t❤✐s
✾
❚❤❡ tr♦✉❜❧❡ ✇✐t❤ tr✉t❤ ❛♥❞ ❢❛❧s✐t② ❛♣♣❡❛rs ❛❧s♦ ✇❤❡♥ ✐t ❝♦♠❡s t♦ ❞❡❛❧ ✇✐t❤ ❜❡❧✐❡❢✿ Pr✐❡st ❬✽✱
♣✳✸✸❪ s✉❣❣❡sts t❤❛t ✏tr✉t❤ ✐s t❤❡ ❛✐♠ ♦❢ ❝♦❣♥✐t✐✈❡ ♣r♦❝❡ss❡s✑✱ ✇❤✐❧❡ ❢❛❧s✐t② ✏❤❛s ♥♦ ✐♥❞❡♣❡♥❞❡♥t
❡♣✐st❡♠♦❧♦❣✐❝❛❧ ❢♦r❝❡✑✳ ❚r✉t❤ ❞♦♠✐♥❛t❡s ❢❛❧s✐t②✱ ❛♥❞ t❤❡r❡ ✐s ♥♦t❤✐♥❣ ❞❛♥❣❡r♦✉s ✐♥ ❜❡❧✐❡✈✐♥❣
s♦♠❡ ❢❛❧s✐t✐❡s✳ ❍♦✇❡✈❡r✱ ❛s Pr✐❡st ❛❧s♦ r❡❝♦❣♥✐③❡s✱ ❢❛❧s✐t② ✐s t❤❡ tr✉t❤ ♦❢ t❤❡ ♥❡❣❛t✐♦♥✳ ❇✉t
t❤❡♥✱ t❤❡ q✉❡st✐♦♥ ❛r✐s❡s✿ ❝❛♥✬t ♥❡❣❛t✐✈❡ st❛t❡♠❡♥ts ❜❡ t❤❡ ❛✐♠ ♦❢ ❝♦❣♥✐t✐✈❡ ♣r♦❝❡ss❡s❄ ❈❛♥✬t
♦♥❡ ❜❡❧✐❡✈❡ ✭✐✳❡✳✱ ❤❛✈❡ ❛s t❤❡ ❛✐♠ ♦❢ ❛ ❝♦❣♥✐t✐✈❡ ♣r♦❝❡ss✮ t❤❛t t❤❡r❡ ✐s ♥♦ ❧❛r❣❡st ♣r✐♠❡ ♥✉♠❜❡r✱
t❤❛t ❇r❛③✐❧ ❞✐❞ ♥♦t ✇✐♥ t❤❡ ✷✵✶✹ ❋■❋❆ ❲♦r❧❞ ❈✉♣✱ ❛♥❞ s♦ ♦♥❄ ■t ✐s ♥♦t ❝❧❡❛r t♦ ✉s ✇❤② s✉❝❤
✐s t❤❡ ❝❛s❡✳ ❚❤♦s❡ ❛r❡ tr✉t❤s t❤❛t ♦♥❡ ♠❛② r❡❛s♦♥❛❜❧② ❜❡❧✐❡✈❡✳ ❍♦✇❡✈❡r✱ t❤✐s s✐t✉❛t✐♦♥ ♦♥❧②
✐❧❧✉str❛t❡s t❤❡ ❦✐♥❞ ♦❢ tr♦✉❜❧❡ t❤❛t ❢❛❧s✐t② ♣♦s❡s ❢♦r ❞✐❛❧❡t❤❡✐sts ❧✐❦❡ Pr✐❡st✳
✶✵✻
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
✇❤♦❧❡ s❡♥t❡♥❝❡ ✐s s✐♠♣❧② ❢❛❧s❡✱ ❞✉❡ t♦ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ♥❡❣❛t✐♦♥ ❛♥❞ ❝♦♥❥✉♥❝t✐♦♥✳
❇✉t ♠♦r❡ ❝❛♥ ❜❡ ♦✛❡r❡❞ ✐♥ t❡r♠s ♦❢ ❡✈✐❞❡♥❝❡ ❢♦r ♦✉r ❝❧❛✐♠✿ ♥♦t❤✐♥❣ ♣r❡✈❡♥ts ✉s
❢r♦♠ ❣♦✐♥❣ ♦♥❡ st❡♣ ❢✉rt❤❡r ✐♥ t❤❡ s✐♠♣❧❡ ▲✐❛r ❛♥❞ s❡♣❛r❛t✐♥❣ t❤❡ ❝♦♥❥✉♥❝t✐♦♥✱
♦❜t❛✐♥✐♥❣ T ✭_¬L^✮ ❛s t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ ❛ ♦♥❡✲st❡♣ ❧♦♥❣❡r ❛r❣✉♠❡♥t✳ ❚❤❛t ✐s✱ t❤❡
❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ s✐♠♣❧❡ ▲✐❛r ♥♦✇ ♦♥❡ st❡♣ ❧♦♥❣❡r ✐s ❛ ❢❛❧s❡ st❛t❡♠❡♥t✱ s✐♠♣❧②
❢❛❧s❡✳ ❚❤❛t ✐s t❤❡ ♠♦st ❞r❛♠❛t✐❝ ❝♦♥❝❧✉s✐♦♥ t❤❛t t❤❡ ❞❡t❛❝❤❛❜❧❡ ❛❝❝♦✉♥t ♦❢
❣❧✉ts ❛✛♦r❞s ✉s✳
❆❣❛✐♥✱ t❤❡ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✐❛♥ ❛❧r❡❛❞② ❦♥❡✇ t❤❛t s✉❝❤ ❛ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ tr✉t❤
❛♥❞ ❢❛❧s✐t② ②✐❡❧❞s ❛ ❢❛❧s✐t②✳ ❇✉t t❤❡ ❞✐❛❧❡t❤❡✐st ✇✐❧❧ ❤❛✈❡ tr♦✉❜❧❡ ❡①♣❧❛✐♥✐♥❣
t❤❛t s♦♠❡t❤✐♥❣ s❤❡ tr❡❛t❡❞ ❛s ❛ s♦✉♥❞ ❛r❣✉♠❡♥t ♥♦✇ ❧❡❛❞s✱ ✇✐t❤ ♦♥❧② ♦♥❡
♠♦r❡ st❡♣✱ t♦ str❛✐❣❤t ❢❛❧s✐t②✳ ❚❤❛t ✐s t❤❡ ♣❛r❛❞✐❣♠ ♦❢ ❛♥ ✐♥✈❛❧✐❞ ❛r❣✉♠❡♥t✳
❚❤❛t ✐s✱ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ▲✐❛r s❡♥t❡♥❝❡ L ❛♥❞ t❤❡ T ✲s❝❤❡♠❛✱ ✇❡ ❝❛♥
❞❡r✐✈❡ ❛ ❢❛❧s✐t②✱ ❛ ♣✉r❡ ❢❛❧s✐t②✳ ■♥ ♠♦❞❡❧ t❤❡♦r❡t✐❝ t❡r♠s✱ ✇❡ ❤❛✈❡ ❛s ♣r❡♠✐s❡s
④ L ↔ F ✭_L^✮✱ T ✲s❝❤❡♠❛⑥✱ ❢r♦♠ ✇❤✐❝❤ ✐t ❢♦❧❧♦✇s F ✭_L^✮✳ ■❢ t❤❡ ♣r❡♠✐s❡s ❛r❡
tr✉❡✱ ❛s t❤❡ ❞✐❛❧❡t❤❡✐sts ❝♦♥t❡♥❞✱ ❛♥❞ t❤❡ r❡❛s♦♥✐♥❣ ✐s s♦✉♥❞✱ ❜✉t t❤❡ ❝♦♥❝❧✉s✐♦♥
✐s ✭❥✉st✮ ❢❛❧s❡✱ ❛s ✐t ❝❧❡❛r❧② ✐s✱ t❤❡♥ ✇❡ ❛r❡ ❢❛❝✐♥❣ ❛♥ ✐♥✈❛❧✐❞ ❛r❣✉♠❡♥t✱ ❛❣❛✐♥
❛❣❛✐♥st t❤❡ ❝❧❛✐♠s ♦❢ t❤❡ ❞✐❛❧❡t❤❡✐sts✳
❍♦✇ t♦ ❡①♣❧❛✐♥ t❤❛t❄ ❚❤❡ ♣♦✐♥t ✐s t❤❛t t❤❡ ❞❡t❛❝❤❛❜❧❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢
❛ ❣❧✉t✱ t❛❦❡♥ ❛s T ✭_L^✮ ∧ T ✭_¬L^✮✱ ✐s ❛ ❝♦♥❥✉♥❝t✐♦♥✳ ❚❤❡ ❞✐❛❧❡t❤❡✐st ❛❝❝❡♣ts
t❤❡ r✉❧❡ ♦❢ s❡♣❛r❛t✐♦♥✱ t❤❛t ✐s✱ ❢r♦♠ α ∧ β ✇❡ ♠❛② ✈❛❧✐❞❧② ❞❡r✐✈❡ ❡❛❝❤ ♦❢ t❤❡
❝♦♥❥✉♥❝ts✳ ❇✉t ❢r♦♠ t❤❡ s❡♥t❡♥❝❡ T ✭_L^✮ ∧ T ✭_¬L^✮ ✇❡ ♠❛② ❞❡r✐✈❡ T ✭_¬L^✮✱ t❤❛t
✐s✱ F ✭_L^✮✳ ❙♦✱ ✐❢ t❤❡ ♣r❡♠✐s❡ ❤❛❞ ❛ ❞❡s✐❣♥❛t❡❞ ✈❛❧✉❡ ✭❣❧✉t ✐s ❛ ❞❡s✐❣♥❛t❡❞ ✈❛❧✉❡
❢♦r t❤❡ ❞✐❛❧❡t❤❡✐st✮✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ ❛ ✈❛❧✐❞ ✐♥❢❡r❡♥❝❡ s❤♦✉❧❞ ❛❧s♦ ❜❡ ❞❡s✐❣♥❛t❡❞✳
❍♦✇❡✈❡r✱ t❤❛t ✐s ❝❧❡❛r❧② ♥♦t t❤❡ ❝❛s❡✦ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s ♦♥❧② ❢❛❧s❡✳ ❚❤❡r❡ ✐s
♦♥❧② ♦♥❡ ♠♦✈❡ t♦ ♣r❡✈❡♥t t❤❡ s❡♣❛r❛t✐♦♥ r✉❧❡ ❢r♦♠ ❜❡✐♥❣ ✐♥✈❛❧✐❞✿ ❛ss✉♠❡ t❤❛t
t❤❡ ♣r❡♠✐s❡ ✐s ❢❛❧s❡ ❛s ✇❡❧❧✳ ❚❤❛t ✐s ✇❤❛t ✇❡ ❤❛✈❡ ❛r❣✉❡❞ ❜❡❢♦r❡✱ ❥♦✐♥✐♥❣ t❤❡
❝❧❛ss✐❝❛❧ ❧♦❣✐❝✐❛♥✳ ❚❤❛t✱ ❤♦✇❡✈❡r✱ ✐s ♥♦t ✇❤❛t t❤❡ ❞✐❛❧❡t❤❡✐st ✇✐s❤❡s✳ ■♥ ❢❛❝t✱
t❤❛t ♠♦✈❡ ✇♦✉❧❞ ❧❡❛❞ t♦ t❤❡ ❝❧❛ss✐❝❛❧ ❝♦♥❝❧✉s✐♦♥ t❤❛t t❤❡ ▲✐❛r ♣❛r❛❞♦① ❤❛s ❛
❢❛❧s❡ ❝♦♥❝❧✉s✐♦♥ ✭❝♦♥tr❛❞✐❝t✐♦♥ ✐♥ t❤❡ ❝❧❛ss✐❝❛❧ s❡♥s❡✮✳
◆♦t✐❝❡ t❤❛t✱ ❛♣❛rt ❢r♦♠ t❤❛t ♣r♦❜❧❡♠ ♦❢ ❣♦✐♥❣ ❢r♦♠ ❛ ❣❧✉t t♦ ❛ ❢❛❧s❡ s❡♥t❡♥❝❡✱
t❤❡r❡ ✐s ❛♥♦t❤❡r ♠❛✐♥ ✇♦rr②✿ t❤❛t ♠♦✈❡ ✐s ✐♠♣♦ss✐❜❧❡ ✐♥ ▲P✱ ✇✐t❤ t❤❡ ✉♥❞❡t❛❝❤✲
❛❜❧❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts✳ ❆ss✉♠✐♥❣ t❤❛t ♦✉r ♣r❡♠✐s❡s ❛r❡ ❣❧✉ts✱ ✇❡ ❝❛♥ ♥❡✈❡r
✈❛❧✐❞❧② ✐♥❢❡r ❛ ❢❛❧s❡ s❡♥t❡♥❝❡✦ ❖❜✈✐♦✉s❧②✱ ✇✐t❤ t❤❛t ❛❝❝♦✉♥t✱ t❤❡ s❡♣❛r❛t✐♦♥ r✉❧❡
✐s s♦✉♥❞ ✐♥ ▲P✳ ❙♦✱ Pr✐❡st✬s ✐♥t✉✐t✐✈❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts ✐♠♣❧②✐♥❣ ❢❛❧s✐t② ❛s ✇❡❧❧ ❛s
tr✉t❤ ✭❬✾✱ ♣✳✾✾❪✮ ❢♦❧❧♦✇s t❤❡ ❞❡t❛❝❤❛❜❧❡ ❛♣♣r♦❛❝❤✱ ❜✉t ✐t ✐s ♥♦t r❡♣r♦❞✉❝✐❜❧❡ ✐♥
▲P✱ ♥♦t ❛s ❛ ✈❛❧✐❞ ✐♥❢❡r❡♥❝❡✱ ❛t ❧❡❛st✳
❖❜✈✐♦✉s❧②✱ t❤❡ ♣r♦❜❧❡♠ ❝♦♠❡s ❛❣❛✐♥ ❢r♦♠ ❛♥ ✐♥❝♦❤❡r❡♥t s✇✐t❝❤ ❜❡t✇❡❡♥ t❤❡
❞❡t❛❝❤❛❜❧❡ ❛♥❞ ✉♥❞❡t❛❝❤❛❜❧❡ ❛❝❝♦✉♥ts ♦❢ ❣❧✉ts✳ ❲❤❡♥ ❣❧✉ts ❛r❡ tr❡❛t❡❞ ❛s ❛
✉♥✐t✱ ❛s ❛ ❦✐♥❞ ♦❢ t❤✐r❞ tr✉t❤ ✈❛❧✉❡ s♦ t❤❛t ♦♥❡ ❝❛♥♥♦t s❡♣❛r❛t❡ tr✉t❤ ❢r♦♠
❢❛❧s✐t②✱ t❤❡♥ t❤❡ ✐♥❢❡r❡♥❝❡ ❢r♦♠ ❣❧✉t t♦ ❢❛❧s✐t② ❝❛♥♥♦t ❜❡ ♠❛❞❡✳ ❍♦✇❡✈❡r✱ t❤❡
✉♥❞❡t❛❝❤❛❜❧❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts ❛❧s♦ ♣r❡✈❡♥ts Pr✐❡st✬s ❛r❣✉♠❡♥ts t❤❛t tr✉t❤ ❤❛s
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
✶✵✼
s♦♠❡ ❦✐♥❞ ♦❢ ❛❞✈❛♥t❛❣❡ ♦✈❡r ❢❛❧s✐t②✿ t❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ s❡♣❛r❛t❡ t❤♦s❡ tr✉t❤
✈❛❧✉❡s ✐♥ ❛ ❣❧✉t✱ s♦ ✐t ❞♦❡s ♥♦t ♠❛❦❡ s❡♥s❡ t♦ s❛② ♦♥❡ ♦❢ t❤❡♠ ❞♦♠✐♥❛t❡s ♦✈❡r
t❤❡ ♦t❤❡r✳ ■♥ t❤❡ ❡♥❞✱ t❤❡ ♣r♦❜❧❡♠ r❡♠❛✐♥s✿ ❤♦✇ s❤♦✉❧❞ ✇❡ ✉♥❞❡rst❛♥❞ ❣❧✉ts
✐♥ ❛ ❝♦❤❡r❡♥t ✇❛②❄
✺ ❈♦♥❝❧✉s✐♦♥
❆s ✇❡ ❤❛✈❡ s❡❡♥✱ t❤❡ ❞✐❛❧❡t❤❡✐st ❤❛s ❛ ❤❛r❞ t✐♠❡ ♠❛❦✐♥❣ t❤❡ ✐❞❡❛ ♦❢ tr✉❡ ❝♦♥✲
tr❛❞✐❝t✐♦♥s ❝♦❤❡r❡♥t✳ ❲❡ ❤❛✈❡ ❡①♣❧♦r❡❞ s♦♠❡ ❞✐✣❝✉❧t q✉❡st✐♦♥s ♣♦s❡❞ t♦ t❤❡
❞✐❛❧❡t❤❡✐st✱ ❛♥❞ t❤❡② ❛❧❧ ♣♦✐♥t t♦ t❤❡ s❛♠❡ ❞✐r❡❝t✐♦♥✿
t❤❡r❡ ✐s s♦♠❡ ❦✐♥❞ ♦❢
❛♠❜✐❣✉✐t② ✐♥ ❞✐❛❧❡t❤❡✐s♠ t❤❛t ♠❛❦❡s ✐t ❛ ❤♦♣❡❧❡ss s♦❧✉t✐♦♥ t♦ t❤❡ ▲✐❛r ♣❛r❛❞♦①
❛♥❞ ❡✈❡♥ ♠♦r❡✱ t❤❛t ♠❛❦❡s ✐t ❤❛r❞ ❢♦r ✉s t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ✐s r❡❛❧❧② ♠❡❛♥t
❜② tr✉❡ ❝♦♥tr❛❞✐❝t✐♦♥s✳ ▲❡t ✉s r❡❝❛❧❧ t❤♦s❡ ❞✐✣❝✉❧t✐❡s✱ r❛t❤❡r ❜r✐❡✢②✳
❚❤❡ ✜rst tr♦✉❜❧❡ ❝♦♠❡s ✇✐t❤ t❤❡ t✇♦ ✈❡rs✐♦♥s ♦❢ t❤❡ ▲✐❛r✱ t❤❡ s✐♠♣❧❡ ▲✐❛r
❛♥❞ t❤❡ ❡①t❡♥❞❡❞ ▲✐❛r✳ ❆s ✇❡ ❤❛✈❡ s❡❡♥✱ ❜② t❤❡ ❞✐❛❧❡t❤❡✐sts✬ ♦✇♥ ❧✐❣❤ts t❤♦s❡ ❛r❡
❞✐st✐♥❝t ♣❛r❛❞♦①❡s✱ ❛♥❞ ✇❤✐❧❡ ✇❡ ♠❛② s❛② ✇❡ ❤❛✈❡ ❛ ❞✐❛❧❡t❤❡✐❛ ✐♥ t❤❡ ❝♦♥❝❧✉s✐♦♥
♦❢ t❤❡ s✐♠♣❧❡ ▲✐❛r✱ t❤❡ s❛♠❡ ❝❛♥♥♦t ❜❡ s❛✐❞ ❛❜♦✉t t❤❡ ❝♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❡①t❡♥❞❡❞
▲✐❛r✳
❚❤❡ ❧❛st ♦♥❡ ✐s ❝❧❡❛r❧② ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉t ✇✐t❤♦✉t ❣❧✉ts ✇❡ ❤❛✈❡ ♥♦
❝❧❡❛r ❛❝❝♦✉♥t ♦❢ ❤♦✇ t♦ ❛✈♦✐❞ ❡①♣❧♦s✐♦♥ ❛♥❞ ❝♦♥s❡q✉❡♥t❧②✱ tr✐✈✐❛❧✐t②✳
❙♦✱ ✐♥
❣❡♥❡r❛❧✱ t❤❡ ❞✐✣❝✉❧t✐❡s ❝♦♠❡ ❢r♦♠ t❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❡①t❡r♥❛❧ ❛♥❞
✐♥t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥s✳ ❲❤✐❧❡ ✐♥t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥s ❞♦ ♣r♦✈✐❞❡ ❢♦r ❣❧✉ts ❛♥❞
✇❡ ❤❛✈❡ ❛ ❧♦❣✐❝❛❧ tr❡❛t♠❡♥t ❢♦r t❤❡♠ ✭✐✳❡✳ t❤❡② ❛r❡ ♥♦t ❡①♣❧♦s✐✈❡✮✱ ❢♦r ❡①t❡r♥❛❧
❝♦♥tr❛❞✐❝t✐♦♥s s✉❝❤ ❛ tr❡❛t♠❡♥t ✐s ❧❛❝❦✐♥❣❀ ✐♥ ❢❛❝t✱ ❡✈❡♥ ❛ tr✉t❤ ✈❛❧✉❡ ❢♦r ❛♥
❡①t❡r♥❛❧ ❝♦♥tr❛❞✐❝t✐♦♥ ✐s ❧❛❝❦✐♥❣✱ ✇❡ s✐♠♣❧② ❞♦ ♥♦t ❦♥♦✇ ✇❤✐❝❤ tr✉t❤ ✈❛❧✉❡ ✇❡
s❤♦✉❧❞ ❛ttr✐❜✉t❡ t♦ ✐t✳
❚❤❡ s❡❝♦♥❞ tr♦✉❜❧❡ ❝♦♠❡s ✇✐t❤ t❤❡ ✈❡r② ❝♦♥❝❡♣t ♦❢ ❣❧✉t ❛♥❞ ✐ts r❡❧❛t✐♦♥
t♦ ❝♦♥tr❛❞✐❝t✐♦♥s ❛♥❞ ❞✐❛❧❡t❤❡✐❛s✳ ❚✇♦ tr❡❛t♠❡♥ts ❢♦r ❣❧✉ts ❛r❡ ❛✈❛✐❧❛❜❧❡✱ ❜♦t❤
❧❡❛❞✐♥❣ t♦ ✈❡r② ❞✐st✐♥❝t r❡s✉❧ts✳ ❚❤❡ ✉♥❞❡t❛❝❤❛❜❧❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts ❞❡❛❧s ✇✐t❤
s❡♥t❡♥❝❡s ❤❛✈✐♥❣ ❛ t❤✐r❞ tr✉t❤ ✈❛❧✉❡✱ t❤❡ ❣❧✉t✱ ✇❤✐❝❤ ♠❡❛♥s ✏tr✉❡ ❛♥❞ ❢❛❧s❡✑✳
❚❤❛t tr❡❛t♠❡♥t ✐s ✜♥❡ ❢♦r t❤❡ ❧♦❣✐❝ ▲P✱ ❛❧t❤♦✉❣❤ ✐t ✈✐♦❧❛t❡s ♦t❤❡r r❡q✉✐r❡♠❡♥ts
♦♥ ❞✐❛❧❡t❤❡✐❛s✱ s✉❝❤ ❛s ❡❝♦♥♦♠② ❛♥❞ r❛r✐t②✳ ❆❧s♦✱ t❤❡② ❧❡❛❞ t♦ ❝♦✉♥t❡r✐♥t✉✐t✐✈❡
r❡s✉❧ts ✇❤❡♥ ❝♦♥❥♦✐♥❡❞ ❜② ❧♦❣✐❝❛❧ ❝♦♥♥❡❝t✐✈❡s ✇✐t❤ ♦t❤❡r ♥♦♥✲❣❧✉tt② s❡♥t❡♥❝❡s❀
✐♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ♠❛② ❤❛✈❡ ❣❧✉tt② s❡♥t❡♥❝❡s t❤❛t ❛r❡ ♥♦t ❝♦♥tr❛❞✐❝t✐♦♥s✳ ❚❤❡
❞❡t❛❝❤❛❜❧❡ ❛❝❝♦✉♥t ♦❢ ❣❧✉ts ✐s ❝❧♦s❡r t♦ t❤❡ tr❡❛t♠❡♥t Pr✐❡st ❣✐✈❡s t♦ ❣❧✉ts ✐♥
✐♥❢♦r♠❛❧ ❞✐s❝✉ss✐♦♥s✱ ❜✉t ✐t ❧❡❛❞s t♦ ❛♥ ❛❜❛♥❞♦♥♠❡♥t ♦❢ ❞✐❛❧❡t❤❡✐s♠✿ ❝♦♥tr❛✲
❞✐❝t✐♦♥s ❛r❡ ❛❧✇❛②s ❢❛❧s❡✱ ❞✉❡ t♦ s✐♠♣❧❡ ❢❛❝ts ♦❢ ❝♦♥❥✉♥❝t✐♦♥ ❛♥❞ tr✉t❤ ✈❛❧✉❡s✳
❆❧s♦✱ t❤❡ s✐♠♣❧❡ r✉❧❡ ♦❢ s❡♣❛r❛t✐♦♥ ♦❢ ❝♦♥❥✉♥❝t✐♦♥ ✐s ✈✐♦❧❛t❡❞ ✐❢ ❣❧✉ts ❛r❡ ♥♦t
t❛❦❡♥ ❛s ❢❛❧s❡ s❡♥t❡♥❝❡s✳
❘❡❝❛❧❧ t❤❛t t❤❡ ❞✐❛❧❡t❤❡✐st ✇❛♥t❡❞ t♦ ♣r♦✈✐❞❡ ❛ ♠♦❞❡❧ ✇❤♦s❡ ♠❛✐♥ ❛✐♠ ✇❛s
t♦ ♣r❡s❡r✈❡ ✐♥t✉✐t✐✈❡ ❢❡❛t✉r❡s ♦❢ ♦✉r ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡ ❛♥❞ t❤❡ ✇♦r❦✐♥❣s ♦❢ s✉❝❤
❧❛♥❣✉❛❣❡✳ ■♥t✉✐t✐✈❡❧②✱ t❤❡ ❝❧❛✐♠ ✐s t❤❛t ♥❛t✉r❛❧ r❡❛s♦♥✐♥❣ ❧❡❛❞s t♦ ❝♦♥tr❛❞✐❝t✐♦♥s
✶✵✽
❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt ❛♥❞ ❊✳ ❙✳ ▼❡❧♦
✇❤❡♥ s❡♠❛♥t✐❝ ❝♦♥❝❡♣ts ❛r❡ ❢r❡❡❧② t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✱ ❛♥❞ t❤❛t ❞❛t❛ s❤♦✉❧❞ ❜❡
♣r❡s❡r✈❡❞ ✐♥ t❤❡ ♠♦❞❡❧✳ ❲❤❛t t❤❡ ❞✐✣❝✉❧t✐❡s ✇❡ ❤❛✈❡ ❡①♣❧♦r❡❞ ❛❧❧ ♣♦✐♥t t♦✱
✐t s❡❡♠s t♦ ✉s✱ ✐s t❤❛t ❛♥♦t❤❡r ♠✉❝❤ ♠♦r❡ ♣♦✇❡r❢✉❧ ✐♥t✉✐t✐♦♥ s❡❡♠s t♦ ❜❡ ✈✐♥✲
❞✐❝❛t❡❞✿ ❝♦♥tr❛❞✐❝t✐♦♥s s❤♦✉❧❞ r❡❛❧❧② ❜❡ ✉♥❞❡rst♦♦❞ ❛s ❛ ✇❛r♥✐♥❣ s✐❣♥❛❧✱ ❛s ❛
s✐❣♥ t❤❛t s♦♠❡t❤✐♥❣ ✐s ❝❧❡❛r❧② ✇r♦♥❣ ✇✐t❤ ♦✉r ❝♦♥❝❡♣t✉❛❧ ♠❛❝❤✐♥❡r②✳ ❘❛t❤❡r
t❤❛♥ ❛❝❝♦♠♠♦❞❛t❡ ❝♦♥tr❛❞✐❝t✐♦♥s✱ ✇❡ s❤♦✉❧❞✱ ♣❡r❤❛♣s✱ s❡❡❦ t♦ ✜♥❞ t❤❡ s♦✉r❝❡
♦❢ t❤❡ ♣r♦❜❧❡♠❀ ✐❢ ✇❡ ❝❛♥✬t ❛❣r❡❡ ♦♥ ✇❤❛t ✐t ✐s✱ ❛t ❧❡❛st ✇❡ ❛❣r❡❡ t❤❛t s♦♠❡t❤✐♥❣
❤❛s ❣♦♥❡ s❡r✐♦✉s❧② ✇r♦♥❣✱ ❛♥❞ t❤✐s✱ ♣❡r❤❛♣s✱ ✐s t❤❡ ♠♦st ❜❛s✐❝ ✐♥t✉✐t✐♦♥✳ ❚❤❡
❝r❡❞❡♥t✐❛❧s ♦❢ t❤✐s ✐♥t✉✐t✐♦♥ s❤♦✉❧❞ ♥♦t ❜❡ ❡❛s✐❧② ❞✐s♠✐ss❡❞✿ ♦✉r ❡✈❡r②❞❛② str✉❣✲
❣❧❡ t♦ ❛✈♦✐❞ ❜❡✐♥❣ ❝❛✉❣❤t ✐♥ ❝♦♥tr❛❞✐❝t✐♦♥ ❛♥❞ ✷✳✵✵✵ ②❡❛rs ♦❢ ❜❛tt❧❡ ❛❣❛✐♥st t❤❡
▲✐❛r ❝❡rt❛✐♥❧② t❡st✐♠♦♥② ✐♥ ✐ts ❢❛✈♦r✳
❆❝❦♥♦✇❧❡❞❣♠❡♥ts
❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤❛♥❦ ❘✐❝❤❛r❞ ❍❡❝❦ ❢♦r ❝♦♠♠❡♥ts ♦♥ ❛ ♣r❡✈✐♦✉s ✈❡rs✐♦♥ ♦❢
t❤✐s ♣❛♣❡r✳
❘❡❢❡r❡♥❝❡s
❬✶❪ ❏✳ ❘✳ ❇✳ ❆r❡♥❤❛rt✳ ▲✐❜❡r❛t✐♥❣ ♣❛r❛❝♦♥s✐st❡♥❝② ❢r♦♠ ❝♦♥tr❛❞✐❝t✐♦♥
❯♥✐✈❡rs❛❧✐s✱ ✈✳✾✱ ♣✳✺✷✸✲✺✹✹✱ ✷✵✶✺✳
▲♦❣✐❝❛
❬✷❪ ❏✳ ❈✳ ❇❡❛❧❧✳ ■♥tr♦❞✉❝t✐♦♥✿ ❛t t❤❡ ■♥t❡rs❡❝t✐♦♥ ♦❢ ❚r✉t❤ ❛♥❞ ❋❛❧s✐t②✳ ■♥✿ ❚❤❡
▲❛✇ ♦❢ ◆♦♥✲❈♦♥tr❛❞✐❝t✐♦♥✿ ◆❡✇ P❤✐❧♦s♦♣❤✐❝❛❧ ❊ss❛②s✳ ❊❞✐t❡❞ ❜② ●✳ Pr✐❡st✱
❏✳ ❈✳ ❇❡❛❧❧ ❛♥❞ ❇✳ ❆r♠♦✉r✲●❛r❜✳❖①❢♦r❞✿ ❈❧❛r❡♥❞♦♥ Pr❡ss✱ ♣✳✶✲✶✾✱ ✷✵✵✹✳
❬✸❪ ❏✳ ✲❨✳ ❇é③✐❛✉✳ ❘♦✉♥❞ ❙q✉❛r❡s ❛r❡ ◆♦ ❈♦♥tr❛❞✐❝t✐♦♥s ✭❚✉t♦r✐❛❧ ♦♥ ◆❡❣❛✲
t✐♦♥✱ ❈♦♥tr❛❞✐❝t✐♦♥✱ ❛♥❞ ❖♣♣♦s✐t✐♦♥✮✳ ■♥✿ ◆❡✇ ❉✐r❡❝t✐♦♥s ✐♥ P❛r❛❝♦♥s✐st❡♥t
▲♦❣✐❝✳ ❊❞✐t❡❞ ❜② ❏✳ ✲❨✳ ❇é③✐❛✉✱ ▼✳ ❈❤❛❦r❛❜♦rt② ❛♥❞ ❙✳ ❉✉tt❛✳ ◆❡✇ ❉❡❧❤✐✿
❙♣r✐♥❣❡r✱ ♣✳✸✾✲✺✺✱ ✷✵✶✺✳
❬✹❪ ❏✳✲❨✳ ❇é③✐❛✉✳ ❚r✐✈✐❛❧ ❞✐❛❧❡t❤❡✐s♠ ❛♥❞ t❤❡ ❧♦❣✐❝ ♦❢ ♣❛r❛❞♦①✳ ▲♦❣✐❝ ❛♥❞ ▲♦❣✐❝❛❧
P❤✐❧♦s♦♣❤②✱ ✈✳✷✺✭✶✮✱ ♣✳✺✶✲✺✻✱ ✷✵✶✻✳
❬✺❪ ●✳ Pr✐❡st✳ ❚❤❡ ❧♦❣✐❝ ♦❢ ♣❛r❛❞♦①✳ ❏♦✉r♥❛❧ ♦❢ P❤✐❧♦s♦♣❤✐❝❛❧ ▲♦❣✐❝✱ ✈✳✽✱ ♣✳✷✶✾✲
✷✹✶✱ ✶✾✼✾✳
❬✻❪ ●✳ Pr✐❡st✳ ▲♦❣✐❝ ♦❢ ♣❛r❛❞♦① r❡✈✐s✐t❡❞✳ ❏♦✉r♥❛❧
♣✳✶✺✸✲✶✼✾✱ ✶✾✽✹✳
♦❢ P❤✐❧♦s♦♣❤✐❝❛❧ ▲♦❣✐❝✱
❬✼❪ ●✳ Pr✐❡st✳ ●❛♣s ❛♥❞ ❣❧✉ts✿ ❛ r❡♣❧② t♦ P❛rs♦♥s✳
❧♦s♦♣❤②✱ ✈✳✷✺✭✶✮✱ ♣✳✺✼✲✻✻✱ ✶✾✾✺✳
✈✳✶✸✱
❈❛♥❛❞✐❛♥ ❏♦✉r♥❛❧ ♦❢ P❤✐✲
✶✵✾
■s ❉✐❛❧❡t❤❡✐s♠ ❙❡❧❢✲❈♦❤❡r❡♥t❄
❬✽❪ ●✳ Pr✐❡st✳ ❲❤❛t✬s ❙♦ ❇❛❞ ❆❜♦✉t ❈♦♥tr❛❞✐❝t✐♦♥s❄ ■♥✿ ❚❤❡ ▲❛✇ ♦❢ ◆♦♥✲
❈♦♥tr❛❞✐❝t✐♦♥✿ ◆❡✇ P❤✐❧♦s♦♣❤✐❝❛❧ ❊ss❛②s✳ ❊❞✐t❡❞ ❜② ●✳ Pr✐❡st✱ ❏✳ ❈✳ ❇❡❛❧❧
❛♥❞ ❇✳ ❆r♠♦✉r✲●❛r❜✳ ❖①❢♦r❞✿ ❈❧❛r❡♥❞♦♥ Pr❡ss✱ ♣✳✷✸✲✸✽✱ ✷✵✵✹✳
❬✾❪ ●✳ Pr✐❡st✳ ■♥ ❈♦♥tr❛❞✐❝t✐♦♥✿ ❛ st✉❞②
❖①❢♦r❞✿ ❖①❢♦r❞ ❯♥✳ Pr❡ss✱ ✷✵✵✻❛✳
❬✶✵❪ ●✳ Pr✐❡st✳
❉♦✉❜t tr✉t❤ t♦ ❜❡ ❛ ❧✐❛r✳
♦❢ t❤❡ tr❛♥s❝♦♥s✐st❡♥t✳
✷♥❞ ❡❞✐t✐♦♥✳
❖①❢♦r❞✿ ❖①❢♦r❞ ❯♥✳ Pr❡ss✱ ✷✵✵✻❜✳
❬✶✶❪ ●✳ Pr✐❡st ❛♥❞ ❋✳ ❇❡rt♦✳ ❉✐❛❧❡t❤❡✐s♠✳ ■♥✿ ❚❤❡ ❙t❛♥❢♦r❞ ❊♥❝②❝❧♦♣❡❞✐❛
♦❢ P❤✐❧♦s♦♣❤②✳ ❙✉♠♠❡r ✷✵✶✸ ❊❞✐t✐♦♥✮✳ ❊❞✐t❡❞ ❜② ❊✳ ◆✳ ❩❛❧t❛✳ ❯❘▲ ❂
❤tt♣✿✴✴♣❧❛t♦✳st❛♥❢♦r❞✳❡❞✉✴❛r❝❤✐✈❡s✴s✉♠✷✵✶✸✴❡♥tr✐❡s✴❞✐❛❧❡t❤❡✐s♠✴✱ ✷✵✶✸✳
❬✶✷❪ ●✳ Pr✐❡st ❛♥❞ ❘✳ ❘♦✉t❧❡②✳ ❚❤❡ ♣❤✐❧♦s♦♣❤✐❝❛❧ s✐❣♥✐✜❝❛♥❝❡ ❛♥❞ ✐♥❡✈✐t❛❜✐❧✐t②
♦❢ ♣❛r❛❝♦♥s✐st❡♥❝②✳ ■♥✿ P❛r❛❝♦♥s✐st❡♥t ▲♦❣✐❝✿ ❊ss❛②s ♦♥ t❤❡ ■♥❝♦♥s✐st❡♥t✳
❊❞✐t❡❞ ❜② ●✳ Pr✐❡st✱ ❘✳ ❘♦✉t❧❡②✱ ❛♥❞ ❏✳ ◆♦r♠❛♥✳ ▼ü♥❝❤❡♥✿ P❤✐❧♦s♦♣❤✐❛
❱❡r❧❛❣✱ ♣✳✹✽✸✕✺✸✾✱ ✶✾✽✾✳
❏♦♥❛s ❘❛❢❛❡❧ ❇❡❝❦❡r ❆r❡♥❤❛rt
❉❡♣❛rt♠❡♥t ♦❢ P❤✐❧♦s♦♣❤②
❋❡❞❡r❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❙❛♥t❛ ❈❛t❛r✐♥❛ ✭❯❋❙❈✮
❋❧♦r✐❛♥ó♣♦❧✐s✱ ❙❈✱ ❇r❛③✐❧
❊✲♠❛✐❧✿ ❥♦♥❛s✳❜❡❝❦❡r✷❅❣♠❛✐❧✳❝♦♠
❊❞❡rs♦♥ ❙❛❢r❛ ▼❡❧♦
❉❡♣❛rt♠❡♥t ♦❢ P❤✐❧♦s♦♣❤②
❋❡❞❡r❛❧ ❯♥✐✈❡rs✐t② ♦❢ ▼❛r❛♥❤ã♦ ✭❯❋▼❆✮
❙ã♦ ▲✉ís✱ ▼❆✱ ❇r❛③✐❧
❊✲♠❛✐❧✿ ❡❞❡rs♦♥s❛❢r❛❅②❛❤♦♦✳❝♦♠✳❜r
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