Formal Veri cation of Cryptographic Protocol Tutorial on BAN Logic

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  ❋♦r♠❛❧ ❱❡r✐✜❝❛t✐♦♥ ♦❢ ❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧ ✫ ❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝

  ❆✈✐♥❛♥t❛ ❚❛r✐❣❛♥

  ❈❡♥t❡r ❢♦r ■♥❢♦r♠❛t✐♦♥ ❙❡❝✉r✐t② ❘❡s❡❛r❝❤

  ●✉♥❛❞❛r♠❛ ❯♥✐✈❡rs✐t②

  ❙❡❝✉r✐t② ✫ ❙❛❢❡t②

  ❙❡❝✉r✐t② ✐s ❢✉♥ ✐❞❡❛ ♦❢ ✿

  ❛tt❛❝❦✐♥❣ ❛♥❞ ❞❡❢❡♥s✐♥❣ ♠❛♥❛❣✐♥❣ ❤✉♠❛♥✬s ♣r♦❜❧❡♠ ✭✉s❡r ✲ ❛tt❛❝❦❡r ✲ ❞❡s✐❣♥❡r✮

  ❙❛❢❡t② ✐s ❛❧s♦ ❢✉♥ ✐❞❡❛ ♦❢ ✿

  ♣r❡✈❡♥t✐♥❣ ❢❛✐❧✉r❡s r❡❛❝t✐♥❣ ✉♣♦♥ ❢❛✐❧✉r❡s ♠❛♥❛❣✐♥❣ ❤✉♠❛♥ ❛♥❞ ❡♥✈✐r♦♥♠❡♥t❛❧

  ❇♦t❤ ✐s ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ ❢❛❝t♦r ♦❢ ❞❡♣❡♥❞❛❜✐❧✐t②

  ❈r②♣t♦❣r❛♣❤② ✭r❡✈✐❡✇✮

  ❆❧❣♦r✐t❤♠ t♦ ♣r♦t❡❝t s❡❝r❡❝② ♦❢ ❞❛t❛ ❆❧s♦ ✉s❡❞ t♦ ❣❛✐♥ ✿

  ❛✉t❤❡♥t✐❝❛t✐♦♥ ✐♥t❡❣r✐t② ♥♦♥ r❡♣✉❞✐❛t✐♦♥

  ■♥❝❧✉❞❡s ✿ ❛❧❣♦r✐t❤♠ ❛♥❞ ❦❡②✭s✮ ■♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ r❡q✉✐r❡s ❛ ♣r♦t♦❝♦❧ ✭❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧✮ ❊①❛♠♣❧❡✿ ❙❙▲✱ ❑❡r❜❡r♦s✱ ❲P❆✱ ❈❍❆P✱ ❈♦♥tr❛❝t s✐❣♥✐♥❣✱ ❊✲✈♦t✐♥❣✱ ❊✲♠♦♥❡②✳

  ❙②♠♠❡tr✐❝ ❈r②♣t✳ A 7→B K

  ✿ ④M ⑥ ❛❜

  Pr✐♥❝✐♣❛❧ A s❡♥❞s B ♠❡ss❛❣❡ M ❡♥❝r②♣t❡❞ ✇✐t❤ s❤❛r❡❞✲❦❡② K ❛❜

  ❑❡② ✐s s❤❛r❡❞ ❜❡t✇❡❡♥ ✷ ♣r✐♥❝✐♣❛❧s

  ✷

  ◆❡❡❞s ◆ ❦❡②s ❢♦r ◆ ♣r✐♥❝✐♣❛❧s ❋❛st ❜✉t ❦❡② ♠❛♥❛❣❡♠❡♥t ✐s ♥♦t ❡❛s②

  ❆s②♠♠❡tr✐❝✴P✉❜❧✐❝ ❑❡② ❈r②♣t✳ A

7→B K

  ✿ ④M ⑥ ❜

  Pr✐♥❝✐♣❛❧ A s❡♥❞s B ♠❡ss❛❣❡ M ❡♥❝r②♣t❡❞ ✇✐t❤ B✬s ♣✉❜❧✐❝✲❦❡② K ❜ ✶

  ❖♥❧② ✇✐t❤ ♣r✐✈❛t❡✲❦❡② K ✱ B ❝❛♥ ❞❡❝r②♣t M

  ❜

  Pr✐♥❝✐♣❛❧ ❤❛s ✐ts ♦✇♥ K ✇❤✐❝❤ ✐s ♣✉❜❧✐s❤❡❞ ❛♥❞ K ✇❤✐❝❤ ♠✉st ❜❡ ❦❡❡♣❡❞ s❡❝r❡t ❑❡② ♠❛♥❛❣❡♠❡♥t ✐s ❧❡ss ❞✐✣❝✉❧t✱ r❡q✉✐r❡s ❈❡rt✐✜❝❛t✐♦♥ ❆✉t❤♦r✐t②

  ❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧

  ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❈r②♣t♦❣r❛♣❤② ❆❧❣♦r✐t❤♠ ❆❝❤✐❡✈✐♥❣ s❡❝✉r✐t② ♣r♦♣❡rt✐❡s ✭❛✉t❤❡♥t✐❝❛t✐♦♥✱ s❡❝r❡❝②✱ ❡t❝✳✮ ❊①❛♠♣❧❡ ✿

  

◆❡❡❞❤❛♠✲❙❝❤r♦❡❞❡r ✭❛✉t❤❡♥t✐❝❛t✐♦♥✮

❑❡r❜❡r♦s ✭❛✉t❤❡♥t✐❝❛t✐♦♥✮ ❙❙▲✴❚▲❙ ✭❛✉t❤ ✲ s❡❝r❡❝②✮

  ❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧

  7→ B ✿ {❑ ❛❜

  

✶} ❑ ❛❜

  7→ B ✿ {◆ ❜

  ▼✺ A

  

❛❜

  7→ A ✿ {◆ ❜ }

  ▼✹ B

  

,

❆} ❑ ❜s

  ▼✸ A

  ❊①❛♠♣❧❡ ✿ ◆❡❡❞❤❛♠✲❙❝❤r♦❡❞❡r Pr♦t♦❝♦❧ ▼✶ A

  } ❑ ❛s

  , ❆} ❑ ❜s

  , { ❑ ❛❜

  , ❇, ❑ ❛❜

  7→ A ✿ {◆ ❛

  ▼✷ S

  7→ S ✿ ❆, ❇, ◆ ❛

  ■♥t♦❞✉❝✐♥❣ ◆♦♥❝❡ (◆)

  ■♥tr♦❞✉❝✐♥❣ ❚✐♠❡❙t❛♠♣ (❚ ) ❛♥❞ ▲✐❢❡t✐♠❡ (▲) ❯s❡❞ ✐♥ ♠❛♥② s②st❡♠✱ ✐♥❝❧✉❞✐♥❣ ❲✐♥❞♦✇s

  } ❑ ❛s

  7→ A ✿ {❚ ❛

  ▼✹ B

  ❑ ❛❜

  ❑ ❜s ✱ {❆, ❚ ❛ }

  

❛❜

, ❆}

  7→ B ✿ {❚ s , ▲, ❑

  ▼✸ A

  , ❆} ❑ ❜s

  ❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧

  , ▲, ❑ ❛❜

  , { ❚ s

  , ▲, ❇, ❑ ❛❜

  7→ A ✿ {❚ s

  ▼✷ S

  7→ S ✿ ❆, ❇

  ▼♦r❡ ❡①❛♠♣❧❡ ✿ ❑❡r❜❡r♦s Pr♦t♦❝♦❧ ▼✶ A

  • ✶} ❑ ❛❜

  ❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧

  Pr♦❜❧❡♠ ✿ ❲r♦♥❣ ❞❡s✐❣♥ ❝♦✉❧❞ ❧❡❛❞ t♦ ✢❛✇

  ◆❡❡❞❤❛♠✲❙❝❤r♦❡❞❡r Pr♦t♦❝♦❧ ❙❙▲✈✶✳✵

  ❲r♦♥❣ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❝♦✉❧❞ ❧❡❛❞ t♦ ✈✉❧♥❡r❛❜✐❧✐t②

  

P❛❞❞✐♥❣ ♣r♦❜❧❡♠ ✐♥ ❙❙▲✱ ❙❙❍✱ ❛♥❞ ❲❚▲❙

  ❱✉❧♥❡r❛❜✐❧✐t② ❛r✐s❡ ❜❡t✇❡❡♥ t✇♦ ♣r♦t❡❝t✐♦♥ t❡❝❤♥♦❧♦❣✐❡s ✭❆♥❞❡rs♦♥✱ ❘♦ss✮

  ❋♦r♠❛❧ ▼❡t❤♦❞ ❢♦r ❈r②♣t✳ Pr♦t♦❝♦❧ ■

  ❋♦r♠❛❧ ▼❡t❤♦❞s ❛r❡ ❛ ♣❛rt✐❝✉❧❛r ❦✐♥❞ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧❧②✲❜❛s❡❞ t❡❝❤♥✐q✉❡s ❢♦r t❤❡ s♣❡❝✐✜❝❛t✐♦♥✱ ❞❡✈❡❧♦♣♠❡♥t ❛♥❞ ✈❡r✐✜❝❛t✐♦♥ ♦❢ ❛❧❣♦r✐t❤♠s✱ s♦❢t✇❛r❡ ❛♥❞ ❤❛r❞✇❛r❡ s②st❡♠s ❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧ ✐s ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ✐s ✉s✐♥❣ ❝r②♣t♦❣r❛♣❤✐❝ ♦♣❡r❛t✐♦♥ t❤❛t ❝❛rr✐❡s ♦✉t s❡❝✉r✐t② ♦❜❥❡❝t✐✈❡✭s✮✱ ✐✳❡✳ ✿

  ❆✉t❤❡♥t✐❝❛t✐♦♥ ❙❡❝r❡❝② ❑❡②✲❊①❝❤❛♥❣❡ ◆♦♥✲❘❡♣✉❞✐❛t✐♦♥

  ■♥ t❤✐s ❝❛s❡ ✐t t♦ ♣r♦✈❡ ❝♦rr❡❝t♥❡ss ✐♥ ❛❝❤✐❡✈✐♥❣ s❡❝✉r✐t② ♣r♦♣❡rt✐❡s t❤❛t t❤❡ ♣r♦t♦❝♦❧ ❝❛rr② ♦✉t✱ ✐✳❡ ✿

  t♦ ♣r♦✈❡ t❤❛t ❑❡r❜❡r♦s ♣r♦t♦❝♦❧ ❞♦❡s ❝♦rr❡❝t ❛✉t❤❡♥t✐❝❛t✐♦♥ ❛s s♣❡❝✐✜❡❞

  ❋♦r♠❛❧ ▼❡t❤♦❞ ❢♦r ❈r②♣t✳ Pr♦t♦❝♦❧ ■■

  ❚❤❡r❡ ❛r❡ t✇♦ ❞❡✈❡❧♦♣♠❡♥t ❛♣♣r♦❛❝❤ ✿

  

❊①t❡♥t✐♦♥ ❢r♦♠ ♠❡t❤♦❞ ✉s❡❞ ✐♥ ❝♦♠♠✉♥✐❝❛t✐♦♥

◆❡✇❧② ❞❡✈❡❧♦♣❡❞ ♠❡t❤♦❞

  ❋♦✉r ❝❧❛ss✐✜❝❛t✐♦♥s ✿ ✶✳ ●❡♥❡r❛❧ ♣✉r♣♦s❡ t♦♦❧s ✸✳ ❊①♣❡rt ❙②st❡♠ ✷✳ ▲♦❣✐❝ ❜❛s❡❞ ✹✳ ❆❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤

  ●❡♥❡r❛❧ P✉r♣♦s❡ ❚♦♦❧s

  ❚r❡❛t❡❞ ❛s ♦r❞✐♥❛r② ❝♦♠♠✉♥✐❝❛t✐♦♥ ♣r♦t♦❝♦❧ ❆❞✈❡rs❛r② ✐s ❡①♣❧✐❝✐t✱ ❝❛♣❛❜❧❡ ✐♥ r❡❛❞✱ ✐♥t❡r❝❡♣t✱ ❛♥❞ ♠♦❞✐❢② ♠❡ss❛❣❡s ▼❡t❤♦❞ ✿ ❋❙▼✱ ❈❙P✱ ❋❉❘✱ P❡tr✐ ◆❡ts ❊①❛♠♣❧❡ ✿ ▲♦t♦s✱ ■♥❛ ❏♦✱ ▼✉r♣❤②

  System State A

  INTRUDER B

  ❯s✐♥❣ ❊①♣❡rt ❙②st❡♠

  ■♥✈❡st✐❣❛t❡ ❡✈❡r② ♣♦ss✐❜❧❡ s❝❡♥❛r✐♦ ♦❢ ❆tt❛❝❦ ✲ ❋❧❛✇ ✲ ❉❡❢❡♥❝❡ ◆❡❡❞s t♦ ❞❡✜♥❡ ✐♥s❡❝✉r❡ st❛t❡s ❛♥❞ s❡❛r❝❤ ♣❛t❤s t♦ t❤❡♠ ▼♦r❡ s✉❝❝❡ss❢✉❧ t❤❛♥ ●❡♥❡r❛❧ P✉r♣♦s❡ ❚♦♦❧s ❊①❛♠♣❧❡ ✿ ■♥t❡rr♦❣❛t♦r ❜② ▼✐❧❧❡♥✱ ◆❘▲ Pr♦t♦❝♦❧ ❆♥❛❧②③❡r ❜② ▼❡❛❞♦✇s✱ ▲♦♥❣❧❡② ❛♥❞ ❘✐❣❜②

  ❆❧❣❡❜r❛✐❝ ❆♣♣r♦❛❝❤

  ❈❛♣❛❜✐❧t✐❡s ✐♥ ♠♦❞❡❧✐♥❣ ❦♥♦✇❧❡❞❣❡ ✇❤✐❝❤ r❡♣r❡s❡♥ts ❝♦♠♣♦♥❡♥t ✐♥ ❝r②♣t♦❣r❛♣❤✐❝ ♦♣❡r❛t✐♦♥ ✭◆♦♥❝❡✱ ❑❡②✭s✮✱ ❛♥❞ ♦❧❞ ♠❡ss❛❣❡s✮ ❊①❛♠♣❧❡ ✿

  ❉♦❧❡✈ ✲ ❨❛♦ ✭t❡r♠ r❡✲✇r✐t✐♥❣ s②st❡♠s✮

❙ − Π ✲ ❈❛❧❝✉❧✉s ❜② ❆❜❛❞✐ ❛♥❞ ●♦r❞♦♥ ✭t♦ ♣r♦✈❡ s❡❝r❡❝②✮

  ▲♦❣✐❝ ❇❛s❡❞

  ❖♥❡ s❡❡s ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦t♦❝♦❧ ❛s ❞✐str✐❜✉t❡❞ ❛❧❣♦r✐t❤♠ ❉❡✈❡❧♦♣ ❧♦❣✐❝s ❢r♦♠ ♠♦❞❛❧ ❧♦❣✐❝ ❚❤❡r❡ ❛r❡ ✐♥❢❡r❡♥❝❡ r✉❧❡s ●♦❛❧ ✐s t♦ ❞❡r✐✈❡❞ st❛t❡♠❡♥ts ✇❤✐❝❤ r❡♣r❡s❡♥ts ❝♦rr❡❝t ❝♦♥❞✐t✐♦♥ ❊①❛♠♣❧❡ ✿ ❇❆◆ ▲♦❣✐❝ ❛♥❞ ●◆❨ ▲♦❣✐❝

  ❇♦♦❦s✱ P❛♣❡rs✱ ❛♥❞ ▲✐♥❦s

  ❙❝❤♥❡✐❞❡r ❡t✳ ❛❧✳ ✏▼♦❞❡❧❧✐♥❣ ❛♥❞ ❆♥❛❧②s✐s ♦❢ ❙❡❝✉r✐t② Pr♦t♦❝♦❧s✑ ❘♦ss ❆♥❞❡rs♦♥✱ ✏❙❡❝✉r✐t② ❊♥❣✐♥❡❡r✐♥❣✑ ❉♦♥❛❧❞ ▼❛❝❦❡♥③✐❡✱ ✏▼❡❝❤❛♥✐③✐♥❣ Pr♦♦❢✑ ▼❛rt✐♥ ❆❜❛❞✐✬s ♣❛♣❡rs ❛t ❤tt♣✿✴✴✇✇✇✳❝s❡✳✉❝s❝✳❡❞✉✴⑦❛❜❛❞✐✴❛❧❧♣❛♣❡rs✳❤t♠❧★❥s❞s P❛♣❡rs ❛t ❤tt♣✿✴✴❝❤❛❝s✳♥r❧✳♥❛✈②✳♠✐❧✴♣r♦❥❡❝ts✴❝r②♣t♦✳❤t♠❧

  ❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ■

  ❇✉rr♦✇s✱ ❆❜❛❞✐✱ ◆❡❡❞❤❛♠✱ ✏❚❤❡ ▲♦❣✐❝ ♦❢ ❆✉t❤❡♥t✐❝❛t✐♦♥✑✱ ❙❘❈ ❘❡s❡❛r❝❤ ❘❡♣♦rt ✸✾✱ ✶✾✽✾✳ ❋✐rst ❛tt❡♠♣t ♦♥ ▲♦❣✐❝ ✐♥ ✈❡r✐❢②✐♥❣ ❆✉t❤❡♥t✐❝❛t✐♦♥ Pr♦t♦❝♦❧s ❉❡✈❡❧♦♣❡❞ ✉s✐♥❣ ❊♣✐st❡♠✐❝ ▲♦❣✐❝ ✭▲♦❣✐❝ ♦❢ ❑♥♦✇❧❡❞❣❡✮ ◆♦t❛t✐♦♥✿

  Pr✐♥❝✐♣❛❧s ✿ P,◗,❘ ❙♣❡❝✐✜❝ Pr✐♥❝✐♣❛❧s ✿ ❆,❇,❙ ❊♥❝r②♣t✐♦♥ ❑❡②s ✿ , ,

  ❙♣❡❝✐✜❝ ❙❤❛r❡❞ ❑❡② ✿ ❑ ❑ ❛s ❑ ❛❜ ❜s , ,

  ❙♣❡❝✐✜❝ P✉❜❧✐❝ ❑❡② ✿ ❑ ❛ ❑ ❑ s − ✶ ✶ − ✶, , ❙♣❡❝✐✜❝ Pr✐✈❛t❡ ❑❡② ✿ ❑ ❑ ❑ ❛ s ❜

  ❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ■■

  ◆♦t❛t✐♦♥✿

  P |≡ ❳ ✿ P ❜❡❧✐❡✈❡s ❳ P ⊳ ❳ ✿ P ❙❡❡s ❳ P |∼ ❳ ✿ P ♦♥❝❡ s❛✐❞ ❳ P Z⇒ ❳ ✿ P ❤❛s ❥✉r✐s❞✐❝t✐♦♥ ♦✈❡r ❳ ♯ ( ❳ ) ✿ ❋♦r♠✉❧❛ ❳ ✐s ❢r❡s❤

  ❑ ←→ P ◗ ✿ P ❛♥❞ ◗ ✉s❡ s❤❛r❡❞✲❦❡② ❑

  ❑ ✶ 7→ P ✿ P ❤❛s ❑ ❛s ❛ ♣✉❜❧✐❝✲❦❡② ❛♥❞ ❝♦rr❡s♣♦♥❞✐♥❣ ❑ ❛s ♣r✐✈❛t❡✲❦❡② ❳ P ◗ ✿ ❋♦r♠✉❧❛ ❳ ✐s ❛ s❡❝r❡t ❦♥♦✇♥ ♦♥❧② t♦ P ❛♥❞ ◗ { ❳ } ❑ ✿ ❋♦r♠✉❧❛ ❳ ❡♥❝r②♣t❡❞ ✉♥❞❡r ❦❡② ❑ h ❳ i ❨ ✿ ❨ ✐s ♣r♦♦❢ ♦❢ ♦r✐❣✐♥ ❢♦r ❳

  ❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ■■■

  P♦st✉❧❛t❡s ▼❡ss❛❣❡ ▼❡❛♥✐♥❣ ✿

  ❑

  ←→ P |≡ ◗ P, P ⊳ {❳ }

  ❑

  ✭✶✮ P |≡ ◗ |∼ ❳

  ❑

  7→ ✶ P |≡ ◗, P ⊳ {❳ }

  ❑

  ✭✷✮ P |≡ ◗ |∼ ❳

  P |≡ ◗ P, P ⊳ h❳ i ❨ ✭✸✮

  P |≡ ◗ |∼ ❳ ◆♦♥❝❡ ✈❡r✐✜❝❛t✐♦♥

  P |≡ ♯(❳ ), P |≡ ◗ |∼ ❳ ✭✹✮

  P |≡ ◗ |≡ ❳

  ❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ■❱

  ❏✉r✐s❞✐❝t✐♦♥ P |≡ ◗ Z⇒ ❳ , P |≡ ◗ |≡ ❳

  ✭✺✮ P |≡ ❳

  ❈♦♥❥✉❝t✐♦♥ ♦❢ ❜❡❧✐❡❢ P |≡ ❳ , P |≡ ❳

  ✭✻✮ P |≡ (❳ ,❨ )

  ❙♦♠❡ ❞❡❝♦♠♣♦s✐t✐♦♥s P |≡ (❳ ,❨ )

  ✭✼✮ P |≡ ❳

  P |≡ ◗ |≡ (❳ ,❨ ) ✭✽✮

  P |≡ ◗ |≡ ❳ P |≡ ◗ |∼ (❳ ,❨ )

  ✭✾✮ P |≡ ◗ |∼ ❳

  ❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ❱

  ▼♦r❡ ❞❡❝♦♠♣♦s✐t✐♦♥s P ⊳ (❳ ,❨ )

  P ⊳ ❳ ✭✶✵✮

  P ⊳ h❳ i

  ❨

  P ⊳ ❳ ✭✶✶✮

  P |≡ ◗

  ❑

  ←→ P, P ⊳ {❳ } ❑

  P ⊳ ❳ ✭✶✷✮

  ♠♦r❡ ♦♥ ❞❡❝♦♠♣♦s✐t✐♦♥s ✳✳✳ P |≡

  ❑

  7→ P, P ⊳ {❳ } ❑ P ⊳ ❳

  ✭✶✸✮

  ❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ❱■ ❑ − ✶

  P |≡ 7→ ◗, P ⊳ {❳ }

  ❑

  ✭✶✹✮ P ⊳ ❳

  ◆♦♥❝❡ ✐♥✢✉❡♥❝❡ P |≡ ♯(❨ )

  ✭✶✺✮ P |≡ ♯(❨ ,❳ )

  ❙❤❛r❡❞✲❑❡② ❈♦♠♠✉t❛t✐✈❡

  ❑

  ←→ P |≡ ❘ ❘

  ✭✶✻✮

❑ ←→

  P |≡ ❘ ❘

  ❑

  P |≡ ◗ |≡ ❘ ←→ ❘ ✭✶✼✮

  ←→ P |≡ ◗ |≡ ❘ ❘

  ❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ❱■■

  ❙❤❛r❡❞✲❙❡❝r❡t ❈♦♠♠✉t❛t✐✈❡ P |≡ ❘

  ❳

  ❘ P |≡ ❘ ′ ❳

  ✭✶✽✮ P |≡ ◗ |≡ ❘

  ❳

  ❘ P |≡ ◗ |≡ ❘ ′ ❳

  ✭✶✾✮

  ❙t❛❣❡s

  ❙t❛❣❡s ✐♥ ✉s✐♥❣ ❇❆◆ ♣r♦♦❢ ✿ ❚r❛♥s❢♦r♠ ♠❡ss❛❣❡ ✐♥t♦ ✐❞❡❛❧✐③❡❞ ❧♦❣✐❝❛❧ ❢♦r♠✉❧❛

  ❙❦✐♣ t❤❡ ♠❡ss❛❣❡ ♣❛rts t❤❛t ❞♦ ♥♦t ❝♦♥tr✐❜✉t❡ t♦ t❤❡ r❡❝❡✐✈❡r✬s ❜❡❧✐❡❢s

  ❙t❛t❡ ❛ss✉♠♣t✐♦♥s ❛❜♦✉t ♦r✐❣✐♥❛❧ ♠❡ss❛❣❡ ▼❛❦❡ ❛♥♥♦t❛t❡❞ ✐❞❡❛❧✐③❡❞ ♣r♦t♦❝♦❧s ❢♦r ❡❛❝❤ ♣r♦t♦❝♦❧ st❛t❡♠❡♥t ✇✐t❤ ❛ss❡rt✐♦♥s

  ❆♣♣❧② ❧♦❣✐❝❛❧ r✉❧❡s t♦ ❛ss✉♠♣t✐♦♥s ❛♥❞ ❛ss❡rt✐♦♥s ❉❡❞✉❝❡ ❜❡❧✐❡❢s ❤❡❧❞ ❛t t❤❡ ❡♥❞ ♦❢ ♣r♦t♦❝♦❧

  ■❞❡❛❧✐③❛t✐♦♥

  ❚♦ ❢♦r♠❛❧✐③❡ ❛♥❞ r❡♠♦✈❡ ❛♠❜✐❣✉✐t② ✐♥ ♣r♦t♦❝♦❧ ❜✐t str✐♥❣ ❙❦✐♣ t❤❡ ♠❡ss❛❣❡ ♣❛rts t❤❛t ❞♦ ♥♦t ❝♦♥tr✐❜✉t❡ t♦ t❤❡ r❡❝❡✐✈❡r✬s ❜❡❧✐❡❢s ❊①❛♠♣❧❡ ✿

  , ❆ → ❇ : {❆,◆ ❑ }

  ❛ ❛❜ ❑ ❛❜

  t♦

  

❛❜

  , ❇ ⊳ {◆ ❆ ←→ ❇}

  

❛ ❑ ❛❜ Pr♦✈✐♥❣ s✐♠♣❧❡ ♣r♦t♦❝♦❧ ■

  ❊①❛♠♣❧❡ ✿ ❖t✇❛② ❘❡❡s Pr♦t♦❝♦❧ ✭✶✾✽✼✮ ,

  ▼✶ ✿ ❆ → ❇ : ▼,❆,❇,{◆ ❛ ▼,❆,❇} ❑ ❛s , , { ,

  ▼✷ ✿ ❇ → ❙ : ▼,❆,❇,{◆ ▼,❆,❇} ◆ ▼,❆,❇}

  ❛ ❑ ❜ ❑ ❛s ❜s

  , , , } { }

  ▼✸ ✿ ❙ → ❇ : ▼,{◆ ❛ ❑ ❛❜ ❑ ◆ ❜ ❑ ❛❜ ❑ ❛s ❜s , }

  ▼✹ ✿ ❇ → ❆ : ▼,{◆ ❑

  ❛ ❛❜ ❑

❛s

  ■❞❡❛❧✐③❡❞ ✐♥t♦ ✿ ,

  ▼✶ ✿ ❆ → ❇ : {◆ ◆ }

  ❛ ❝ ❑ ❛s

  , , , } { }

  ▼✷ ✿ ❇ → ❙ : {◆ ❛ ◆ ❝ ❑ ◆ ◆ ❝ ❑ ❛s ❜s ❜ ▼✸ ✿

  ❑ ❑ ❛❜ ❛❜

  , ←→ } , { , ←→ } ❙ → ❇ : {◆ ❛ ❆ ❇,❇ |∼ ◆ ❝ ◆ ❆ ❇,❆ |∼ ◆ ❛

  ❑ ❜ ❑ ❛s ❜s❛❜

  , ←→ }

  ▼✹ ✿ ❇ → ❆ : {◆ ❛ ❆ ❇,❇ |∼ ◆ ❝ ❑ ❛s Pr♦✈✐♥❣ s✐♠♣❧❡ ♣r♦t♦❝♦❧ ■■

  ❲❡ ❤❛✈❡ ❛ss✉♠♣t✐♦♥s ✿ ❆ |≡ ❆

  ←→ ❇ ❆ |≡ ❙ Z⇒ ❇ |∼ ❳

  ←→ ❇

  ❑

  ←→ ❙ ❆ |≡ ❙ Z⇒ ❆

  ❑ ❛s

  ) ❙ |≡ ❆

  ❝

  ) ❆ |≡ ♯(◆

  ❛

  ❆ |≡ ♯(◆

  ❑

  ❑ ❛s

  ❇ |≡ ❙ Z⇒ ❆

  ←→ ❇

  ❑ ❛❜

  ←→ ❙ ❙ |≡ ❆

  ❑ ❜s

  ❙ |≡ ❇

  ←→ ❙

  ❑ ❜s

  ←→ ❙ ❇ |≡ ❇

  ❇ |≡ ❙ Z⇒ ❆ |∼ ❳ ❇ |≡ ♯(◆ ❜ ) Pr♦✈✐♥❣ s✐♠♣❧❡ ♣r♦t♦❝♦❧ ■■■

  P❘❖❖❋ ❲❡ ❜❡❣✐♥ ✇✐t❤ ▼✷

  } ❑ ❜s

  , ◆ ❝ )

  ←→ ❙ ✶ ❙ |≡ ❇ |∼ (◆ ❜

  

◆ ❝

} ❑ ❜s ❙ |≡ ❇ ❑ ❜s

  ❜✳ ❙ ⊳ {◆ ❜ ,

  ❝ )

  ❛ , ◆

  ←→ ❙ ✶ ❙ |≡ ❆ |∼ (◆

  

◆ ❝

} ❑ ❛s ❙ |≡ ❆ ❑ ❛s

  ✭❜✮ ❛✳ ❙ ⊳ {◆ ❛ ,

  , ◆ ❝

  ❙ ⊳ {◆ ❛

  ✭❛✮ ❙ ⊳ {◆ ❜

  } ❑ ❛s

  , ◆ ❝

  ✶✵ ❙ ⊳ {◆ ❛

  

}

❜s

  , ◆ ❝

  , { ◆ ❜

  } ❑ ❛s

  , ◆ ❝

  ❛♥❞ t❤❡♥ ▼✸ Pr♦✈✐♥❣ s✐♠♣❧❡ ♣r♦t♦❝♦❧ ■❱ ❑ ❑ ❛❜ ❜s

  , ←→ } ←→

  ❇ ⊳ {◆ ❜ ❆ ❇,❆ |∼ ◆ ❝ ❑ ❙ |≡ ❇ ❙ ❜s ✶ ❑ ❛❜

  , ←→ ) ) ❇ |≡ ❙ |∼ (◆ ❜ ❆ ❇,❆ |∼ ◆ ❝ ❇ |≡ ♯(◆ ❜ ✶✺

  ❑ ❛❜ ←→ ) ❇ |≡ ♯(❆ ❇,❆ |∼ ◆

  ❝ ✹

  ❑ ❛❜ ←→ ) ❇ |≡ ❙ |≡ (❆ ❇,❆ |∼ ◆ ❝

  ✼ ❑ ❛❜

  ←→ ❇ |≡ ❙ |≡ ❆ ❇ ✭❛✮ ❇ |≡ ❙ |≡ ❆ |∼ ◆ ❝ ✭❜✮

  ❢r♦♠ ▼✸

  ❑ ❑ ❛❜ ←→ ←→

❇ |≡ ❙ |≡ ❆ ❇ ❇ |≡ ❙ Z⇒ ❆ ❇

  ❛✳ ✺

  ❑ ❛❜ ←→

❇ |≡ ❆ ❇

  

❇ |≡ ❙ |≡ ❆ |∼ ◆ ❝ ❇ |≡ ❙ Z⇒ ❆ |∼ ❳

❜✳

  ✺

❇ |≡ ❆ |∼ ◆ ❝

  ❛❢t❡r t❤❛t ▼✹ ✿ Pr♦✈✐♥❣ s✐♠♣❧❡ ♣r♦t♦❝♦❧ ❱ ❆ ⊳ {◆ ❛

  , ❆

  ←→ ❇,❇ |∼ ◆ ❝ ) ✼

  ❆ |≡ ❙ |≡ ❇ |∼ ◆ ❝ ❆ |≡ ❙ Z⇒ ❇ |∼ ❳ ❆ |≡ ❇ |∼ ◆ ❝ ❆ |≡ ♯(◆ ❝ ) ✹

  ←→ ❇ ❜✳

  ✺ ❆ |≡ ❆ ❑ ❛❜

  ❑ ←→ ❇

  ❑ ❛❜ ←→ ❇ ❆ |≡ ❙ Z⇒ ❆

  ❛✳ ❆ |≡ ❙ |≡ ❆

  ❢r♦♠ ▼✹

  ←→ ❇ ✭❛✮ ❆ |≡ ❙ |≡ ❇ |∼ ◆ ❝ ✭❜✮

  ❆ |≡ ❙ |≡ ❆ ❑ ❛❜

  ❆ |≡ ❙ |≡ (❆ ❑ ❛❜

  ❑ ❛❜ ←→

  ❝ ) ✹

  ❑ ❛❜ ←→ ❇,❇ |∼ ◆

  ) ✶✺ ❆ |≡ ♯(❆

  ←→

❇,❇ |∼ ◆ ❝

) ❆ |≡ ♯(◆ ❛

  , ❆ ❑ ❛❜

  ❙ ✶ ❆ |≡ ❙ |∼ (◆ ❛

  ❑ ❛s ←→

  ❑ ❛s ❆ |≡ ❆

  ❇,❇ |∼ ◆ ❝ }

  ❆ |≡ ❇ |≡ ◆ ❝

  ▲✐♠✐t❛t✐♦♥s

  ❇❆◆ ❞♦❡s ♥♦t ❝❛t❝❤ ❛❧❧ ♣r♦t♦❝♦❧ ✢❛✇s

  ❋❛❧s❡✲♣♦s✐t✐✈❡s ❝❛♥ r❡s✉❧t

  ❆ ♣r✐♥❝✐♣❛❧✬s ❜❡❧✐❡❢s ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❛t ❧❛t❡r st❛❣❡s ♦❢ t❤❡ ♣r♦t♦❝♦❧

  ◆♦ ❞✐✈✐s✐♦♥ ♦❢ t✐♠❡ ✐♥ ♣r♦t♦❝♦❧ r✉♥

  Pr♦✈✐❞❡s ❛ ♣r♦♦❢ ♦❢ tr✉st ♦♥ ♣❛rt ♦❢ ♣r✐♥❝✐♣❧❡s✱ ❜✉t ♥♦t ❛ ♣r♦♦❢ ♦❢ s❡❝✉r✐t②

  

❋✐♥❛❧ ❜❡❧✐❡❢s ❝❛♥ ❜❡ ❜❡❧✐❡✈❡❞ ♦♥❧② ✐❢ ❛❧❧ ♦r✐❣✐♥❛❧ ❛ss✉♠♣t✐♦♥s

❤♦❧❞ tr✉❡

  ❇❆◆ ❞♦❡s ♥♦t ❛❝❝♦✉♥t ❢♦r ✐♠♣r♦♣❡r ❡♥❝r②♣t✐♦♥

  ❚❤❡ ❊♥❞

  ❚❤❛♥❦ ❨♦✉ ❱✐❡❧❡♥ ❉❛♥❦

  ❚❛❝❦ så ▼②❝❦❡t ▼❛t✉r ◆✉✇✉♥

  ❚❡r✐♠❛❦❛s✐❤

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