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rts tst r s st t tst s st
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❆ r❡s❡❛r❝❤ ♣♦✐♥t ♦❢ ✈✐❡✇ ❆♥t♦✐♥❡ ❘♦❧❧❡t
■◆P ❇♦r❞❡❛✉① ✲ ❯♥✐✈❡rs✐té ❞❡ ❇♦r❞❡❛✉① ✲ ▲❛❇❘■ ✭❯▼❘ ❈◆❘❙ ✺✽✵✵✮✱ ❋r❛♥❝❡ r♦❧❧❡t❅❧❛❜r✐✳❢r ❤tt♣✿✴✴✇✇✇✳❧❛❜r✐✳❢r✴⑦r♦❧❧❡t
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❤✐s t❛❧❦ s❤♦✉❧❞ ✿ Pr♦✈✐❞❡ s♦♠❡ ❜❛s✐❝s ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❡st✐♥❣ Pr❡♣❛r❡ t❤❡ ❛✉❞✐❡♥❝❡ ❢♦r ❢♦❧❧♦✇✐♥❣ ♣r❡s❡♥t❛t✐♦♥s Pr♦✈✐❞❡ s♦♠❡ ❤✐st♦r✐❝❛❧ r❡s❡❛r❝❤ r❡s✉❧ts ♦♥ t❡st✐♥❣ ❚❤✐s t❛❧❦ ✐s ♥♦t ✿ ▼② ♣❡rs♦♥❛❧ r❡s❡❛r❝❤ r❡s✉❧ts❛ ❆♥ ❡①❤❛✉st✐✈❡ ♣r❡s❡♥t❛t✐♦♥ ❆❞✈❛♥❝❡❞ r❡s❡❛r❝❤✱ ♠♦r❡ ❛ ❣❡♥❡r❛❧ ✈✐❡✇ ♦♥ t❤❡ t♦♣✐❝
❛✐♠♣❧②✐♥❣ t❤❛t ■ ❛♠ ♥♦t ❛ s♣❡❝✐❛❧✐st ♦❢ ❛❧❧ ♦❢ t❤❡ ♣r❡s❡♥t❡❞ t♦♣✐❝s✦ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✶ ●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ✷ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ✸ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣
✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡s ✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▲❛❜❡❧❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s
✹ ❈♦♥❝❧✉s✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✶ ●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ✷ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ✸ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣
✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡s ✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▲❛❜❡❧❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s
✹ ❈♦♥❝❧✉s✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋❛♠♦✉s ❜✉❣s ✭♥♦♥ ❡①❤❛✉st✐✈❡ ♣❛♥♦r❛♠❛✮ ❆r✐❛♥❡ ✺✳✵✶ ✭✶✾✾✻✮ P❛tr✐♦t ♠✐ss✐❧❡ ✭✶✾✾✶✮ ❋✐rst P❡♥t✐✉♠ ❈❤✐♣ ✭✶✾✾✹✮ ❚❤❡r❛❝ ✷✺ ✭✶✾✽✺✲✶✾✽✼✮ ✳✳✳ ✭❧♦♥❣ ❧♦♥❣ ❧✐st✮ ❯r❜❛♥ ❧❡❣❡♥❞s ❛❧s♦ ✭❋✶✻ ✜❣❤t❡r ❥❡t ❜✉❣✮ ❲♦♥❞❡r✐♥❣ ✇❤❛t t❤❡ ❝♦st ♦❢ s♦❢t✇❛r❡ ❜✉❣s❄ → ✩ ✸✶✷ ❇✐❧❧✐♦♥s ♣❡r ②❡❛r ❛❝❝♦r❞✐♥❣ t♦ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② ✭✷✵✶✸✮✳ ■♥ ❢❛❝t ✐t ❞❡♣❡♥❞s ♦♥ ❤♦✇ ❧❛t❡ ②♦✉ ✜♥❞ ✐t✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋❛♠♦✉s ❜✉❣s ✭♥♦♥ ❡①❤❛✉st✐✈❡ ♣❛♥♦r❛♠❛✮ ❆r✐❛♥❡ ✺✳✵✶ ✭✶✾✾✻✮ P❛tr✐♦t ♠✐ss✐❧❡ ✭✶✾✾✶✮ ❋✐rst P❡♥t✐✉♠ ❈❤✐♣ ✭✶✾✾✹✮ ❚❤❡r❛❝ ✷✺ ✭✶✾✽✺✲✶✾✽✼✮ ✳✳✳ ✭❧♦♥❣ ❧♦♥❣ ❧✐st✮ ❯r❜❛♥ ❧❡❣❡♥❞s ❛❧s♦ ✭❋✶✻ ✜❣❤t❡r ❥❡t ❜✉❣✮ ❲♦♥❞❡r✐♥❣ ✇❤❛t t❤❡ ❝♦st ♦❢ s♦❢t✇❛r❡ ❜✉❣s❄ → ✩ ✸✶✷ ❇✐❧❧✐♦♥s ♣❡r ②❡❛r ❛❝❝♦r❞✐♥❣ t♦ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② ✭✷✵✶✸✮✳ ■♥ ❢❛❝t ✐t ❞❡♣❡♥❞s ♦♥ ❤♦✇ ❧❛t❡ ②♦✉ ✜♥❞ ✐t✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❙♦✉r❝❡ ✿ ■❇▼ ❙②st❡♠s ❙❝✐❡♥❝❡s ■♥st✐t✉t❡
❈♦♥s❡q✉❡♥❝❡s ✿ Pr♦❞✉❝t r❡❝❛❧❧ ✭P❡♥t✐✉♠ ❈❤✐♣✱ ❚♦②♦t❛ ❜r❛❦❡ s②st❡♠ ❜✉❣ ✭✷✵✵✾✮✮ ❙♦♠❡t✐♠❡s ❧♦ss ♦❢ ❤✉♠❛♥ ❧✐❢❡s ✭❚❤❡r❛❝ ✷✺✱ P❛tr✐♦t✮✱ ❧♦ss ♦❢ ❡①♣❡♥s✐✈❡ s②st❡♠ ✭❆r✐❛♥❡ ✺✳✵✶✮ ❙♣❛❝❡ ❞♦♠❛✐♥ ✿ s❡♥❞ ♣❛t❝❤❡s ✭◆❆❙❆ ❈✉r✐♦s✐t② Pr♦❜❡ ♦♥ ▼❛rs✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❉②♥❛♠✐❝ t❡st✐♥❣ ❉②♥❛♠✐❝ t❡st✐♥❣ ✿ t❤❡ s♦❢t✇❛r❡ ✭■❯❚✮❛ ✐s ❡①❡❝✉t❡❞ ✐♥ ♦r❞❡r ❚♦ ❡♥s✉r❡ ❛ ✏❝♦rr❡❝t✑ ❜❡❤❛✈✐♦✉r ❚♦ ✜♥❞ ❜✉❣s ❛♥❞ ❞❡❢❛✉❧ts ✭▼②❡rs✮ ✭= ❙t❛t✐❝ t❡st✐♥❣✮ → t❤✐s ♣r❡s❡♥t❛t✐♦♥ ✇✐❧❧ ❢♦❝✉s ♠❛✐♥❧② ♦♥ ❞②♥❛♠✐❝ t❡st✐♥❣ t❡❝❤♥✐q✉❡s
❛■♠♣❧❡♠❡♥t❛t✐♦♥ ❯♥❞❡r ❚❡st
❜✉t ♥♦t s♦ s✐♠♣❧❡ ✳✳✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✽✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❡st✐♥❣ ✐s ❛ ❞✐✣❝✉❧t✴❡①♣❡♥s✐✈❡ t❛s❦ ❇✳ ●❛t❡s ✿ ✏✺✵✪ ♦❢ t❤❡ ♣❡♦♣❧❡ ❛t ▼✐❝r♦s♦❢t ❛r❡ t❡st❡rs✱ ❛♥❞ t❤❡ ♣r♦❣r❛♠♠❡rs s♣❡♥❞ ✺✵✪ ♦❢ t❤❡✐r t✐♠❡ t❡st✐♥❣✱ t❤✉s ▼✐❝r♦s♦❢t ✐s ♠♦r❡ ♦❢ ❛ t❡st✐♥❣ t❤❛♥ ❛ ❞❡✈❡❧♦♣♠❡♥t ♦r❣❛♥✐③❛t✐♦♥✑❛
❛❤tt♣✿✴✴✇✇✇✳✐♥❢♦r♠❛t✐♦♥✇❡❡❦✳❝♦♠✴st♦r②✴■❲❑✷✵✵✷✵✺✶✼❙✵✵✶✶
= ⇒ ■♠♣♦rt❛♥t r❡s❡❛r❝❤ ❞♦♠❛✐♥ ■❞❡❛❧❧② ❛ t❡st s❤♦✉❧❞ ❜❡ ❡①❤❛✉st✐✈❡✱ ❜✉t ♥♦t ♣♦ss✐❜❧❡ ✐♥ ♣r❛❝t✐❝❡✳✳✳ ❆ s✐♠♣❧❡ ❢✉♥❝t✐♦♥
✐♥t ♣r♦❞✉❝t✭✐♥t ✐✱ ✐♥t ❥✮❀
264 ♣♦ss✐❜✐❧✐t✐❡s✳ ❈♦♥s✐❞❡r✐♥❣ ♦♥❡ t❡st ♣❡r ♠✐❝r♦✲s❡❝♦♥❞ → ✺✽✸✵✵✵ ②❡❛rs✳✳✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❡st✐♥❣ ✐s ❛ ❞✐✣❝✉❧t✴❡①♣❡♥s✐✈❡ t❛s❦ ❇✳ ●❛t❡s ✿ ✏✺✵✪ ♦❢ t❤❡ ♣❡♦♣❧❡ ❛t ▼✐❝r♦s♦❢t ❛r❡ t❡st❡rs✱ ❛♥❞ t❤❡ ♣r♦❣r❛♠♠❡rs s♣❡♥❞ ✺✵✪ ♦❢ t❤❡✐r t✐♠❡ t❡st✐♥❣✱ t❤✉s ▼✐❝r♦s♦❢t ✐s ♠♦r❡ ♦❢ ❛ t❡st✐♥❣ t❤❛♥ ❛ ❞❡✈❡❧♦♣♠❡♥t ♦r❣❛♥✐③❛t✐♦♥✑❛
❛❤tt♣✿✴✴✇✇✇✳✐♥❢♦r♠❛t✐♦♥✇❡❡❦✳❝♦♠✴st♦r②✴■❲❑✷✵✵✷✵✺✶✼❙✵✵✶✶
= ⇒ ■♠♣♦rt❛♥t r❡s❡❛r❝❤ ❞♦♠❛✐♥ ■❞❡❛❧❧② ❛ t❡st s❤♦✉❧❞ ❜❡ ❡①❤❛✉st✐✈❡✱ ❜✉t ♥♦t ♣♦ss✐❜❧❡ ✐♥ ♣r❛❝t✐❝❡✳✳✳ ❆ s✐♠♣❧❡ ❢✉♥❝t✐♦♥
✐♥t ♣r♦❞✉❝t✭✐♥t ✐✱ ✐♥t ❥✮❀
264 ♣♦ss✐❜✐❧✐t✐❡s✳ ❈♦♥s✐❞❡r✐♥❣ ♦♥❡ t❡st ♣❡r ♠✐❝r♦✲s❡❝♦♥❞ → ✺✽✸✵✵✵ ②❡❛rs✳✳✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❉✐❥❦str❛ ✿ ✏❚❡st✐♥❣ s❤♦✇s t❤❡ ♣r❡s❡♥❝❡✱ ♥♦t t❤❡ ❛❜s❡♥❝❡ ♦❢ ❜✉❣s ✏ = ⇒ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❡st✐♥❣ ✐s t♦ ✐♥❝r❡❛s❡ ❝♦♥✜❞❡♥❝❡ ✐♥ t❤❡ s②st❡♠ ✭■❯❚✮ ▼❛✐♥ ♣r♦❜❧❡♠s ✿
❋✐♥❞ ❛ ✏r❡♣r❡s❡♥t❛t✐✈❡✑ s❛♠♣❧❡ ♦❢ ❞❛t❛ ✭❚❡st ❉❛t❛ ✭❚❉✮✮✱ ♣r♦✈✐❞✐♥❣ ✏❡♥♦✉❣❤✑ ❝♦♥✜❞❡♥❝❡ ❆✉t♦♠❛t✐❝❛❧❧② ❣❡♥❡r❛t❡ t❤✐s s❛♠♣❧❡ ♦❢ ❞❛t❛ ❆✉t♦♠❛t✐❝❛❧❧② ♣r♦✈✐❞❡ ❛ ✈❡r❞✐❝t ✭♦r❛❝❧❡ ♣r♦❜❧❡♠✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❉✐❥❦str❛ ✿ ✏❚❡st✐♥❣ s❤♦✇s t❤❡ ♣r❡s❡♥❝❡✱ ♥♦t t❤❡ ❛❜s❡♥❝❡ ♦❢ ❜✉❣s ✏ = ⇒ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❡st✐♥❣ ✐s t♦ ✐♥❝r❡❛s❡ ❝♦♥✜❞❡♥❝❡ ✐♥ t❤❡ s②st❡♠ ✭■❯❚✮ ▼❛✐♥ ♣r♦❜❧❡♠s ✿
❋✐♥❞ ❛ ✏r❡♣r❡s❡♥t❛t✐✈❡✑ s❛♠♣❧❡ ♦❢ ❞❛t❛ ✭❚❡st ❉❛t❛ ✭❚❉✮✮✱ ♣r♦✈✐❞✐♥❣ ✏❡♥♦✉❣❤✑ ❝♦♥✜❞❡♥❝❡ ❆✉t♦♠❛t✐❝❛❧❧② ❣❡♥❡r❛t❡ t❤✐s s❛♠♣❧❡ ♦❢ ❞❛t❛ ❆✉t♦♠❛t✐❝❛❧❧② ♣r♦✈✐❞❡ ❛ ✈❡r❞✐❝t ✭♦r❛❝❧❡ ♣r♦❜❧❡♠✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❤r❡❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❡st✐♥❣ ✭❚r❡t♠❛♥s✮ ▲❡✈❡❧ ♦❢ ❞❡t❛✐❧ ✿ ❯♥✐t✱ ▼♦❞✉❧❡✱ ■♥t❡❣r❛t✐♦♥ ✳✳✳ ❆❝❝❡ss✐❜✐❧✐t② ✿ ❲❤✐t❡✲❜♦①✱ ❜❧❛❝❦✲❜♦①✳ ❈❤❛r❛❝t❡r✐st✐❝s ✿ ❈♦♥❢♦r♠❛♥❝❡✱ ❘♦❜✉st♥❡ss✱ P❡r❢♦r♠❛♥❝❡✱ ✳✳✳ → t❤✐s ♣r❡s❡♥t❛t✐♦♥ ✇✐❧❧ ❢♦❝✉s ♠❛✐♥❧② ♦♥ ✭✉♥✐t✮ ❝♦♥❢♦r♠❛♥❝❡ t❡st✐♥❣ ❚✇♦ ✭❝♦♠♣❧❡♠❡♥t❛r②✮ ♠❛✐♥ ❛♣♣r♦❛❝❤❡s ♦❢ ❝♦♥❢♦r♠❛♥❝❡ t❡st✐♥❣ ✿ ❋✉♥❝t✐♦♥❛❧ ❚❡st✐♥❣ ✿ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❙②st❡♠ ❯♥❞❡r ❚❡st ✭❙❯❚✮✳ ■❢ s♣❡❝✐✜❝❛t✐♦♥ ❂ ▼♦❞❡❧ → ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❙tr✉❝t✉r❛❧ ❚❡st✐♥❣ ✿ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ ✏str✉❝t✉r❡✑ ♦❢ t❤❡ ❙❯❚✱ ❣❡♥❡r❛❧❧② t❤❡ ❙♦✉r❝❡ ❈♦❞❡ ✭❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❤r❡❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❡st✐♥❣ ✭❚r❡t♠❛♥s✮ ▲❡✈❡❧ ♦❢ ❞❡t❛✐❧ ✿ ❯♥✐t✱ ▼♦❞✉❧❡✱ ■♥t❡❣r❛t✐♦♥ ✳✳✳ ❆❝❝❡ss✐❜✐❧✐t② ✿ ❲❤✐t❡✲❜♦①✱ ❜❧❛❝❦✲❜♦①✳ ❈❤❛r❛❝t❡r✐st✐❝s ✿ ❈♦♥❢♦r♠❛♥❝❡✱ ❘♦❜✉st♥❡ss✱ P❡r❢♦r♠❛♥❝❡✱ ✳✳✳ → t❤✐s ♣r❡s❡♥t❛t✐♦♥ ✇✐❧❧ ❢♦❝✉s ♠❛✐♥❧② ♦♥ ✭✉♥✐t✮ ❝♦♥❢♦r♠❛♥❝❡ t❡st✐♥❣ ❚✇♦ ✭❝♦♠♣❧❡♠❡♥t❛r②✮ ♠❛✐♥ ❛♣♣r♦❛❝❤❡s ♦❢ ❝♦♥❢♦r♠❛♥❝❡ t❡st✐♥❣ ✿ ❋✉♥❝t✐♦♥❛❧ ❚❡st✐♥❣ ✿ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❙②st❡♠ ❯♥❞❡r ❚❡st ✭❙❯❚✮✳ ■❢ s♣❡❝✐✜❝❛t✐♦♥ ❂ ▼♦❞❡❧ → ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❙tr✉❝t✉r❛❧ ❚❡st✐♥❣ ✿ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ ✏str✉❝t✉r❡✑ ♦❢ t❤❡ ❙❯❚✱ ❣❡♥❡r❛❧❧② t❤❡ ❙♦✉r❝❡ ❈♦❞❡ ✭❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣✮ → ✐♥ ♠❛♥② ❜♦♦❦s✱ ❢✉♥❝t✐♦♥❛❧ ❂ ❜❧❛❝❦✲❜♦① ❀ str✉❝t✉r❛❧ ❂ ✇❤✐t❡✲❜♦①
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❤r❡❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❡st✐♥❣ ✭❚r❡t♠❛♥s✮ ▲❡✈❡❧ ♦❢ ❞❡t❛✐❧ ✿ ❯♥✐t✱ ▼♦❞✉❧❡✱ ■♥t❡❣r❛t✐♦♥ ✳✳✳ ❆❝❝❡ss✐❜✐❧✐t② ✿ ❲❤✐t❡✲❜♦①✱ ❜❧❛❝❦✲❜♦①✳ ❈❤❛r❛❝t❡r✐st✐❝s ✿ ❈♦♥❢♦r♠❛♥❝❡✱ ❘♦❜✉st♥❡ss✱ P❡r❢♦r♠❛♥❝❡✱ ✳✳✳ → t❤✐s ♣r❡s❡♥t❛t✐♦♥ ✇✐❧❧ ❢♦❝✉s ♠❛✐♥❧② ♦♥ ✭✉♥✐t✮ ❝♦♥❢♦r♠❛♥❝❡ t❡st✐♥❣ ❚✇♦ ✭❝♦♠♣❧❡♠❡♥t❛r②✮ ♠❛✐♥ ❛♣♣r♦❛❝❤❡s ♦❢ ❝♦♥❢♦r♠❛♥❝❡ t❡st✐♥❣ ✿ ❋✉♥❝t✐♦♥❛❧ ❚❡st✐♥❣ ✿ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❙②st❡♠ ❯♥❞❡r ❚❡st ✭❙❯❚✮✳ ■❢ s♣❡❝✐✜❝❛t✐♦♥ ❂ ▼♦❞❡❧ → ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❙tr✉❝t✉r❛❧ ❚❡st✐♥❣ ✿ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ ✏str✉❝t✉r❡✑ ♦❢ t❤❡ ❙❯❚✱ ❣❡♥❡r❛❧❧② t❤❡ ❙♦✉r❝❡ ❈♦❞❡ ✭❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣✮ → ✐♥ ❛♥② ❝❛s❡✱ ❚❉ ❛r❡ ❛♣♣❧✐❡❞ ♦♥ t❤❡ ■❯❚ ❛♥❞ r❡s✉❧t ✐s ❝♦♠♣❛r❡❞ t♦ t❤❡ s♣❡❝✐✜❝❛t✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■♥t❡❣r❛t✐♦♥ ✿ ❝♦♠❜✐♥✐♥❣ ❛❧r❡❛❞② t❡st❡❞ ❝♦♠♣♦♥❡♥ts✳
❊✈❡♥ ✐❢ ❡❛❝❤ ❝♦♠♣♦♥❡♥t ✐s ✇♦r❦✐♥❣ ✜♥❡ → ✐♥t❡❣r❛t✐♦♥ ♠❛②
r❡✈❡❛❧ ♥❡✇ ❜✉❣s ▼❛✐♥ ✭❢✉♥❝t✐♦♥❛❧✲❞❡❝♦♠♣♦s✐t✐♦♥ ❜❛s❡❞✮ str❛t❡❣✐❡s ❇✐❣✲❜❛♥❣ → ✐♥t❡❣r❛t❡ ❛❧❧ ❝♦♠♣♦♥❡♥ts t♦❣❡t❤❡r✱ t❤❡♥ t❡st t❤❡ ✇❤♦❧❡ ❇♦tt♦♠✲✉♣ → ❢r♦♠ ❧❡❛✈❡s t♦ r♦♦t ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ tr❡❡ ❚♦♣✲❞♦✇♥ → ❢r♦♠ r♦♦t t♦ ❧❡❛✈❡s ♦❢ t❤❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ tr❡❡ → ♥❡❡❞ t♦ ✉s❡ st✉❜s ❙❛♥❞✇✐❝❤ → ❝♦♠❜✐♥✐♥❣ ❇♦tt♦♠✲✉♣ ❛♥❞ ❚♦♣✲❉♦✇♥ ❖t❤❡r ❛♣♣r♦❛❝❤❡s ♠❛② ❜❡ ✉s❡❞ ✭❈❛❧❧✲❣r❛♣❤ ❜❛s❡❞✱ P❛t❤ ❜❛s❡❞✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋✉♥❝t✐♦♥❛❧ ❚❡st✐♥❣ ❈❤❡❝❦✐♥❣ t❤❛t t❤❡ ■❯❚ ♠❡❡ts t❤❡ ❢✉♥❝t✐♦♥❛❧ r❡q✉✐r❡♠❡♥ts✳ ❉✐✈✐❞❡❞ ✐♥t♦ ❢♦✉r ❝♦♠♣♦♥❡♥ts ✿ ❯♥✐t✱ ■♥t❡❣r❛t✐♦♥✱ ❙②st❡♠✱ ❆❝❝❡♣t❛♥❝❡ ◆♦♥✲❋✉♥❝t✐♦♥❛❧ ❚❡st✐♥❣ ❚❡st✐♥❣ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ❛❣❛✐♥st ♥♦♥✲❢✉♥❝t✐♦♥❛❧ r❡q✉✐r❡♠❡♥ts ✿ P❡r❢♦r♠❛♥❝❡✱ ▲♦❛❞✱ ❙tr❡ss✱ ❙❡❝✉r✐t②✱ ✳✳✳ ❚❤❡ ♣r❡✈✐♦✉s ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐❧❧ ❜❡ ✉s❡❞ ✐♥ t❤✐s ♣r❡s❡♥t❛t✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋✉♥❝t✐♦♥❛❧ ❚❡st✐♥❣ ❈❤❡❝❦✐♥❣ t❤❛t t❤❡ ■❯❚ ♠❡❡ts t❤❡ ❢✉♥❝t✐♦♥❛❧ r❡q✉✐r❡♠❡♥ts✳ ❉✐✈✐❞❡❞ ✐♥t♦ ❢♦✉r ❝♦♠♣♦♥❡♥ts ✿ ❯♥✐t✱ ■♥t❡❣r❛t✐♦♥✱ ❙②st❡♠✱ ❆❝❝❡♣t❛♥❝❡ ◆♦♥✲❋✉♥❝t✐♦♥❛❧ ❚❡st✐♥❣ ❚❡st✐♥❣ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ❛❣❛✐♥st ♥♦♥✲❢✉♥❝t✐♦♥❛❧ r❡q✉✐r❡♠❡♥ts ✿ P❡r❢♦r♠❛♥❝❡✱ ▲♦❛❞✱ ❙tr❡ss✱ ❙❡❝✉r✐t②✱ ✳✳✳ ❚❤❡ ♣r❡✈✐♦✉s ❝❧❛ss✐✜❝❛t✐♦♥ ✇✐❧❧ ❜❡ ✉s❡❞ ✐♥ t❤✐s ♣r❡s❡♥t❛t✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
▼✉t❛t✐♦♥ ❚❡st✐♥❣ ✏❚❡st✐♥❣ t❤❡ ❚❡st❡r✑ ❆♣♣❧② t✐♥② ♠✉t❛t✐♦♥s ♦♥ t❤❡ ❙❯❚ ✭✉s✉❛❧❧② ♦♥ t❤❡ s♦✉r❝❡ ❝♦❞❡✮ ❈❤❡❝❦ t❤❛t t❤❡ t❡st ❝❛s❡s ✏❦✐❧❧ t❤❡ ♠✉t❛♥ts✑ → ♠✉t❛t✐♦♥ s❝♦r❡ ❉✐✣❝✉❧t② ✿ ❛♣♣❧② s✐❣♥✐✜❝❛♥t ♠✉t❛t✐♦♥s❀ ❡q✉✐✈❛❧❡♥t ♠✉t❛♥ts ✭♥♦♥✲✮❘❡❣r❡ss✐♦♥ ❚❡st✐♥❣ ❱❡r✐❢②✐♥❣ t❤❛t ❛♥ ✉♣❞❛t❡ ♦❢ t❤❡ ❙❯❚ ❞♦❡s ♥♦t ❛✛❡❝t ♦t❤❡r ♣❛rts ❈❤❡❝❦ t❤❛t ♦❧❞❡r t❡st ❝❛s❡s st✐❧❧ ♣❛ss ❯s✉❛❧❧② ❜❛s❡❞ ♦♥ ❢✉♥❝t✐♦♥❛❧ t❡st ❝❛s❡s ❉✐✣❝✉❧t② ✿ ❝♦st❧②✱ ✜♥❞ ❛ s✉❜s❡t ♦❢ t❡st ❝❛s❡s s✉✐t❡❞ ❢♦r r❡❣r❡ss✐♦♥ t❡st✐♥❣ ❘❡♠❛r❦ ✿ s♦♠❡t✐♠❡s✱ ❛ ❞✐st✐♥❝t✐♦♥ ✐s ♠❛❞❡ ❜❡t✇❡❡♥ r❡❣r❡ss✐♦♥ ❛♥❞ ♥♦♥✲r❡❣r❡ss✐♦♥ t❡st✐♥❣
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
▼✉t❛t✐♦♥ ❚❡st✐♥❣ ✏❚❡st✐♥❣ t❤❡ ❚❡st❡r✑ ❆♣♣❧② t✐♥② ♠✉t❛t✐♦♥s ♦♥ t❤❡ ❙❯❚ ✭✉s✉❛❧❧② ♦♥ t❤❡ s♦✉r❝❡ ❝♦❞❡✮ ❈❤❡❝❦ t❤❛t t❤❡ t❡st ❝❛s❡s ✏❦✐❧❧ t❤❡ ♠✉t❛♥ts✑ → ♠✉t❛t✐♦♥ s❝♦r❡ ❉✐✣❝✉❧t② ✿ ❛♣♣❧② s✐❣♥✐✜❝❛♥t ♠✉t❛t✐♦♥s❀ ❡q✉✐✈❛❧❡♥t ♠✉t❛♥ts ✭♥♦♥✲✮❘❡❣r❡ss✐♦♥ ❚❡st✐♥❣ ❱❡r✐❢②✐♥❣ t❤❛t ❛♥ ✉♣❞❛t❡ ♦❢ t❤❡ ❙❯❚ ❞♦❡s ♥♦t ❛✛❡❝t ♦t❤❡r ♣❛rts ❈❤❡❝❦ t❤❛t ♦❧❞❡r t❡st ❝❛s❡s st✐❧❧ ♣❛ss ❯s✉❛❧❧② ❜❛s❡❞ ♦♥ ❢✉♥❝t✐♦♥❛❧ t❡st ❝❛s❡s ❉✐✣❝✉❧t② ✿ ❝♦st❧②✱ ✜♥❞ ❛ s✉❜s❡t ♦❢ t❡st ❝❛s❡s s✉✐t❡❞ ❢♦r r❡❣r❡ss✐♦♥ t❡st✐♥❣ ❘❡♠❛r❦ ✿ s♦♠❡t✐♠❡s✱ ❛ ❞✐st✐♥❝t✐♦♥ ✐s ♠❛❞❡ ❜❡t✇❡❡♥ r❡❣r❡ss✐♦♥ ❛♥❞ ♥♦♥✲r❡❣r❡ss✐♦♥ t❡st✐♥❣
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✶ ●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ✷ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ✸ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣
✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡s ✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▲❛❜❡❧❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s
✹ ❈♦♥❝❧✉s✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
P❛rt ❡ss❡♥t✐❛❧❧② ❜❛s❡❞ ♦♥ ✿ ❬❇❍✵✾❪ ❙✳ ❇❛r❞✐♥ ❛♥❞ P✳ ❍❡rr♠❛♥✱ ✏Pr✉♥✐♥❣ t❤❡ ❙❡❛r❝❤ ❙♣❛❝❡ ✐♥ P❛t❤✲❜❛s❡❞ ❚❡st ●❡♥❡r❛t✐♦♥✑✱ ■❈❙❚ ✵✾ ❬●♦t✶✵❪ ❆✳ ●♦t❧✐❡❜✱ ✏❈♦♥str❛✐♥t✲❇❛s❡❞ t❡st✐♥❣✑✱ ♣r❡s❡♥t❛t✐♦♥ ❛t ❯♣♣s❛❧❛ ❯♥✐✈❡rs✐t②✱ ▼❛② ✷✵✶✵✳ ❬❘✉❡✵✽❪ ▼✳ ❘✉❡❤❡r✱ ❙♦❢t✇❛r❡ t❡st✐♥❣ ❝♦✉rs❡s✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ ❙♦✉r❝❡ ❈♦❞❡ ♦❢ t❤❡ ■❯❚ ■❞❡❛❧❧②✱ t❤❡ ❜❡st ❚❉ ✇♦✉❧❞ ❝♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ ❡①❡❝✉t✐♦♥s✳ ❇✉t ✉s✉❛❧❧② ♥♦t ♣♦ss✐❜❧❡ ✐♥ ♣r❛❝t✐❝❡✳ ❚❤❡ ♠♦r❡ ✇❡ ❝♦✈❡r ❝♦❞❡✱ t❤❡ ♠♦r❡ ❝♦♥✜❞❡♥t ✇❡ ❛r❡✱ ❜✉t ❚❤❡ ♠♦r❡ ✇❡ ❝♦✈❡r ❝♦❞❡✱ t❤❡ ♠♦r❡ ❚❉ ✇❡ ♥❡❡❞ t♦ ❣❡♥❡r❛t❡ ❛♥❞ ❛♣♣❧② ◆♦t✐♦♥ ♦❢ ❝♦✈❡r❛❣❡ ❝r✐t❡r✐♦♥ ❚❤❡r❡ ❡①✐sts ❛♥ ♦r❞❡r✐♥❣ ❜❡t✇❡❡♥ ❝♦✈❡r❛❣❡ ❝r✐t❡r✐❛ ✿ ❛❧❧ st❛t❡♠❡♥ts < ✳✳✳ < ❛❧❧ ❡①❡❝✉t✐♦♥s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❈❋● ❉✐r❡❝t❡❞ ●r❛♣❤ r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣♦ss✐❜❧❡ ♣❛t❤s ♦❢ t❤❡ ♣r♦❣r❛♠ ❇✉✐❧t ❢r♦♠ t❤❡ s♦✉r❝❡ ❝♦❞❡ ❆ t❡st ♠❛② ❜❡ s❡❡♥ ❛s ❛ ♣❛t❤ ✐♥ t❤❡ ❈❋● ❉✐r❡❝t ❧✐♥❦ ❜❡t✇❡❡♥ ❝♦❞❡ ❝♦✈❡r❛❣❡ ❛♥❞ ❈❋● ❝♦✈❡r❛❣❡ ❊❛s② t♦ ♦❜t❛✐♥✱ ❡✳❣✳ ✇✐t❤ ❣❝❝ ✿
❣❝❝ ✲❢❞✉♠♣✲tr❡❡✲❝❢❣ ✳✳✳
◆♦t ❛♥ ❡q✉✐✈❛❧❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ✿ → r✐s❦ ♦❢ ❛❞❞✐♥❣ ✉♥❢❡❛s✐❜❧❡ ♣❛t❤s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✽✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❈❋● ❉✐r❡❝t❡❞ ●r❛♣❤ r❡♣r❡s❡♥t✐♥❣ t❤❡ ♣♦ss✐❜❧❡ ♣❛t❤s ♦❢ t❤❡ ♣r♦❣r❛♠ ❇✉✐❧t ❢r♦♠ t❤❡ s♦✉r❝❡ ❝♦❞❡ ❆ t❡st ♠❛② ❜❡ s❡❡♥ ❛s ❛ ♣❛t❤ ✐♥ t❤❡ ❈❋● ❉✐r❡❝t ❧✐♥❦ ❜❡t✇❡❡♥ ❝♦❞❡ ❝♦✈❡r❛❣❡ ❛♥❞ ❈❋● ❝♦✈❡r❛❣❡ ❊❛s② t♦ ♦❜t❛✐♥✱ ❡✳❣✳ ✇✐t❤ ❣❝❝ ✿
❣❝❝ ✲❢❞✉♠♣✲tr❡❡✲❝❢❣ ✳✳✳
◆♦t ❛♥ ❡q✉✐✈❛❧❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ✿ → r✐s❦ ♦❢ ❛❞❞✐♥❣ ✉♥❢❡❛s✐❜❧❡ ♣❛t❤s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✽✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✴✴ ♣r♦❞✉❝t ♦❢ t✇♦ ✐♥t ✐♥t ♣r♦❞✭✐♥t ✐✱ ✐♥t ❥✮ ④ ✐♥t ❦❂✵❀ ✐❢ ✭✐ ❂❂ ✷✮ ❦ ❂ ✐❁❁✶❀ ✴✴ ❡rr♦r ❤❡r❡❀ s❤♦✉❧❞ ❜❡ ❥❁❁✶ ❡❧s❡ ④ ✇❤✐❧❡ ✭✐❃✵✮ ④ ❦ ❂ ❦✰❥❀ ✐✲✲❀ ⑥ ⑥ r❡t✉r♥ ❦❀ ⑥ ❦❂✵ ❦❂✐❁❁✶ r❡t✉r♥ ❦ ❦❂❦✰❥ ✐❂✐✲✶ ✐❂❂✷ ✐✦❂✷ ✐❁❂✵ ✐❃✵
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❈♦✈❡r❛❣❡ ❝r✐t❡r✐❛ ❤✐❡r❛r❝❤② ✭♥♦t ❡①❤❛✉st✐✈❡✮ ✿ ❆❧❧ st❛t❡♠❡♥ts ✭❚❊❘✶✮ ❂ ❆❧❧ ♥♦❞❡s ♦❢ t❤❡ ❈❋● ❆❧❧ ❞❡❝✐s✐♦♥s ✭❚❊❘✷✮ ❂ ❆❧❧ ❜r❛♥❝❤❡s ♦❢ t❤❡ ❈❋● ✳✳✳ ❆❧❧ ❝♦♥❞✐t✐♦♥s ✭❇❈❈❈✱ ❇r❛♥❝❤ ❈♦♥❞✐t✐♦♥ ❈♦♠❜✐♥❛t✐♦♥ ❈♦✈❡r❛❣❡✮ ✿ ❡❛❝❤ ❛t♦♠✐❝ ♣r❡❞✐❝❛t❡ ✭✐✳❡✳ ❝♦♥❞✐t✐♦♥✮ ✐s t❡st❡❞ ✇✐t❤ ❛ tr✉❡ ✈❛❧✉❡ ❛♥❞ ❛ ❢❛❧s❡ ✈❛❧✉❡ ▼❈❉❈ ✭♠♦❞✐✜❡❞ ❝♦♥❞✐t✐♦♥ ✴ ❞❡❝✐s✐♦♥ ❝♦✈❡r❛❣❡✮ ❈❤❡❝❦ ✏t❤❡ r♦❧❡✑ ♦❢ ❡❛❝❤ ❝♦♥❞✐t✐♦♥ ✐♥ t❤❡ ❞❡❝✐s✐♦♥ ✳✳✳ ❆❧❧ ✐✲♣❛t❤s ✭❲❤❡♥ ❢❡❛s✐❜❧❡✱✮ ❧♦♦♣ j t✐♠❡s ✐♥ ❡❛❝❤ ❧♦♦♣ ✭0 ≤ j ≤ i✮✳ ❆❧❧ ❡①❡❝✉t✐♦♥s ❂ ❆❧❧ ✭❢❡❛s✐❜❧❡✮ ♣❛t❤s → ❯s✉❛❧❧② ✐♥✜♥✐t❡ ❖t❤❡r ♣♦ss✐❜❧❡ ❛♣♣r♦❛❝❤❡s ✿ ❡✳❣✳ ❉❛t❛ ❋❧♦✇ ❜❛s❡❞
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❈♦✈❡r❛❣❡ ❝r✐t❡r✐❛ ❤✐❡r❛r❝❤② ✭♥♦t ❡①❤❛✉st✐✈❡✮ ✿ ❆❧❧ st❛t❡♠❡♥ts ✭❚❊❘✶✮ ❂ ❆❧❧ ♥♦❞❡s ♦❢ t❤❡ ❈❋● ❆❧❧ ❞❡❝✐s✐♦♥s ✭❚❊❘✷✮ ❂ ❆❧❧ ❜r❛♥❝❤❡s ♦❢ t❤❡ ❈❋● ✳✳✳ ❆❧❧ ❝♦♥❞✐t✐♦♥s ✭❇❈❈❈✱ ❇r❛♥❝❤ ❈♦♥❞✐t✐♦♥ ❈♦♠❜✐♥❛t✐♦♥ ❈♦✈❡r❛❣❡✮ ✿ ❡❛❝❤ ❛t♦♠✐❝ ♣r❡❞✐❝❛t❡ ✭✐✳❡✳ ❝♦♥❞✐t✐♦♥✮ ✐s t❡st❡❞ ✇✐t❤ ❛ tr✉❡ ✈❛❧✉❡ ❛♥❞ ❛ ❢❛❧s❡ ✈❛❧✉❡ ▼❈❉❈ ✭♠♦❞✐✜❡❞ ❝♦♥❞✐t✐♦♥ ✴ ❞❡❝✐s✐♦♥ ❝♦✈❡r❛❣❡✮ ❈❤❡❝❦ ✏t❤❡ r♦❧❡✑ ♦❢ ❡❛❝❤ ❝♦♥❞✐t✐♦♥ ✐♥ t❤❡ ❞❡❝✐s✐♦♥ ✳✳✳ ❆❧❧ ✐✲♣❛t❤s ✭❲❤❡♥ ❢❡❛s✐❜❧❡✱✮ ❧♦♦♣ j t✐♠❡s ✐♥ ❡❛❝❤ ❧♦♦♣ ✭0 ≤ j ≤ i✮✳ ❆❧❧ ❡①❡❝✉t✐♦♥s ❂ ❆❧❧ ✭❢❡❛s✐❜❧❡✮ ♣❛t❤s → ❯s✉❛❧❧② ✐♥✜♥✐t❡ ❖t❤❡r ♣♦ss✐❜❧❡ ❛♣♣r♦❛❝❤❡s ✿ ❡✳❣✳ ❉❛t❛ ❋❧♦✇ ❜❛s❡❞
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❈♦♥str❛✐♥t✲❇❛s❡❞ ❚❡st✐♥❣ ✭❈❇❚✮ ❈❇❚ ✐s t❤❡ ♣r♦❝❡ss ♦❢ ❣❡♥❡r❛t✐♥❣ t❡st ❝❛s❡s ❛❣❛✐♥st ❛ t❡st✐♥❣ ♦❜❥❡❝t✐✈❡ ❜② ✉s✐♥❣ ❝♦♥str❛✐♥t s♦❧✈✐♥❣ t❡❝❤♥✐q✉❡s ✭●♦t❧✐❡❜✮ Pr✐♥❝✐♣❧❡ ♦❢ ❚❡st ●❡♥❡r❛t✐♦♥
❣❡♥❡r❛t❡ ❛ ❚❉ t❤❛t r❡❛❝❤❡s t❤✐s ❧♦❝❛t✐♦♥ ❚r❛♥s❢♦r♠ ✭♣❛rt ♦❢✮ t❤❡ ♣r♦❣r❛♠ ✐♥t♦ ❛ ❧♦❣✐❝❛❧ ❢♦r♠✉❧❛ ϕ✱ s✉❝❤ t❤❛t s♦❧✈✐♥❣ ϕ ♣r♦✈✐❞❡s ❛ ❚❉ → ❉♦❡s ♥♦t s♦❧✈❡ t❤❡ ♦r❛❝❧❡ ♣r♦❜❧❡♠✳ → P♦✐♥t❡rs ♠❛② ❧❡❛❞ t♦ ❞✐✣❝✉❧t ♣r♦❜❧❡♠s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❈♦♥str❛✐♥t✲❇❛s❡❞ ❚❡st✐♥❣ ✭❈❇❚✮ ❈❇❚ ✐s t❤❡ ♣r♦❝❡ss ♦❢ ❣❡♥❡r❛t✐♥❣ t❡st ❝❛s❡s ❛❣❛✐♥st ❛ t❡st✐♥❣ ♦❜❥❡❝t✐✈❡ ❜② ✉s✐♥❣ ❝♦♥str❛✐♥t s♦❧✈✐♥❣ t❡❝❤♥✐q✉❡s ✭●♦t❧✐❡❜✮ Pr✐♥❝✐♣❧❡ ♦❢ ❚❡st ●❡♥❡r❛t✐♦♥
❣❡♥❡r❛t❡ ❛ ❚❉ t❤❛t r❡❛❝❤❡s t❤✐s ❧♦❝❛t✐♦♥ ❚r❛♥s❢♦r♠ ✭♣❛rt ♦❢✮ t❤❡ ♣r♦❣r❛♠ ✐♥t♦ ❛ ❧♦❣✐❝❛❧ ❢♦r♠✉❧❛ ϕ✱ s✉❝❤ t❤❛t s♦❧✈✐♥❣ ϕ ♣r♦✈✐❞❡s ❛ ❚❉ → ❉♦❡s ♥♦t s♦❧✈❡ t❤❡ ♦r❛❝❧❡ ♣r♦❜❧❡♠✳ → P♦✐♥t❡rs ♠❛② ❧❡❛❞ t♦ ❞✐✣❝✉❧t ♣r♦❜❧❡♠s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
P❛t❤ ♣r❡❞✐❝❛t❡
Π ✐❢ ❢♦r ❛ ❣✐✈❡♥ s❡t ♦❢ ✈❛❧✉❡s V ✱ V | = ϕΠ = ⇒ t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ t❤❡ ♣r♦❣r❛♠ ♦♥ V ❢♦❧❧♦✇s Π → ❋✐♥❞ ❛ s♦❧✉t✐♦♥ ✭✐❢ ❛♥②✮ t♦ ϕΠ ✐♥ ♦r❞❡r t♦ ❛❝t✐✈❛t❡ Π ◆❡❡❞ t♦ r❡♠❡♠❜❡r t❤❡ ✈❛❧✉❡s ♦❢ ✈❛r✐❛❜❧❡s ❛❧♦♥❣ t❤❡ ♣❛t❤ → ♥❡❡❞ t♦ r❡♥❛♠❡ ❡❛❝❤ ✈❛r✐❛❜❧❡ ✐♥ ❝❛s❡ ♦❢ ❛ss✐❣♥♠❡♥t → ❙❙❆✶ ❢♦r♠ ❘❡♠❛r❦ ✿ ❯s✐♥❣ ❣❝❝✱ ❙❙❆ ❢♦r♠ ❝❛♥ ❜❡ ❡❛s✐❧② ♦❜t❛✐♥❡❞ ✿
❣❝❝ ✲❙ ✲❢❞✉♠♣✲tr❡❡✲ss❛ ✳✳✳
✶✭❙✐♥❣❧❡ ❙t❛t✐❝ ❆ss✐❣♥♠❡♥t✮ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✶ r❡❛❞ ✭②✱③✮❀ ✲❃ ②✵✱ ③✵ ❛s ✐♥♣✉ts ✷ ② ❂ ② ✰ ✷❀ ✲❃ ②✶ ❂ ②✵ ✰ ✷ ✸ ① ❂ ② ✰ ✹❀ ✲❃ ①✶ ❂ ②✶ ✰ ✹ ✹ ✐❢ ✭① ❃ ✷ ✯ ③✮ ✲❃ ①✶ ❃ ✷ ✯ ③✵ ♦r ①✶ ❁❂ ✷ ✯ ③✵ ❞❡♣❡♥❞✐♥❣ ♦♥ ♣❛t❤ ✺ ① ❂ ② ✰ ✷❀ ✲❃ ①✷ ❂ ②✶ ✰ ✷ ❋♦r t❤❡ ♣❛t❤ ✿ 1 → 2 → 3 → (4, true) → 5✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r❡❞✐❝❛t❡ ✐s ✿ y1 = y0 + 2 ∧ x1 = y1 + 4 ∧ x1 > 2 ∗ z0 ∧ x2 = y1 + 2✳ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✐♥♣✉ts✱ ✇❡ ❤❛✈❡ y0 + 6 > 2 ∗ z0 ❊①❛♠♣❧❡ ♦❢ ❚❉ ✿ y = 0, z = 0
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✶ r❡❛❞ ✭②✱③✮❀ ✲❃ ②✵✱ ③✵ ❛s ✐♥♣✉ts ✷ ② ❂ ② ✰ ✷❀ ✲❃ ②✶ ❂ ②✵ ✰ ✷ ✸ ① ❂ ② ✰ ✹❀ ✲❃ ①✶ ❂ ②✶ ✰ ✹ ✹ ✐❢ ✭① ❃ ✷ ✯ ③✮ ✲❃ ①✶ ❃ ✷ ✯ ③✵ ♦r ①✶ ❁❂ ✷ ✯ ③✵ ❞❡♣❡♥❞✐♥❣ ♦♥ ♣❛t❤ ✺ ① ❂ ② ✰ ✷❀ ✲❃ ①✷ ❂ ②✶ ✰ ✷ ❋♦r t❤❡ ♣❛t❤ ✿ 1 → 2 → 3 → (4, true) → 5✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r❡❞✐❝❛t❡ ✐s ✿ y1 = y0 + 2 ∧ x1 = y1 + 4 ∧ x1 > 2 ∗ z0 ∧ x2 = y1 + 2✳ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✐♥♣✉ts✱ ✇❡ ❤❛✈❡ y0 + 6 > 2 ∗ z0 ❊①❛♠♣❧❡ ♦❢ ❚❉ ✿ y = 0, z = 0
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✐❢ ✭❛❃✵✮ ✲❃ ✐❢ ✭❛✵ ❃ ✵✮ ✐❂✹❀ ✲❃ ✐✶❂✹❀ ❡❧s❡ ✲❃ ❡❧s❡ ✐❂✺❀ ✲❃ ✐✷❂✺❀ ✲❃ ✐✸❂♣❤✐✭✐✶ ✱✐✷✮❀ ✐❢ ✭❛✵ ❃ ✵✮ ✐✶❂✹ ✐✷❂✺ ✐✸❂♣❤✐✭✐✶✱✐✷✮❀
Φ✲❡①♣r → ❞❡❝✐❞❡ t❤❡ ✈❛❧✉❡ ♦❢ i3 ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣❛t❤ ✉s❡❞ t♦ r❡❛❝❤ ✐t a0 > 0 = ⇒ i1 = 4 ∧ i3 = i1 ¬(a0 > 0) = ⇒ i2 = 5 ∧ i3 = i2 ❈❤♦✐❝❡ ♦❢ ♣❛t❤ ✭❥♦✐♥ ♦♣❡r❛t♦r✮ ❥♦✐♥(a0 > 0 ∧ i1 = 4 ∧ i3 = i1, ¬(a0 > 0) ∧ i2 = 5 ∧ i3 = i2)
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❲❤✐❧❡ ✿ Φ✲❡①♣r ❛❞❞❡❞ ❥✉st ❜❡❢♦r❡ t❤❡ ❞❡❝✐s✐♦♥
■♥✐t✐❛❧ ❝♦❞❡
① ❂ ✶❀ ✇❤✐❧❡ ✭ ① ✦❂ ✶✵ ✮ ④ ❝ ❂ ①❀ ① ❂ ① ✰ ✶❀ ⑥ ♣r✐♥t✭①✮❀
❙❙❆ ❝♦❞❡
①✶ ❂ ✶❀ ①✷ ❂ ♣❤✐✭①✶ ✱ ①✸✮❀ ✇❤✐❧❡ ✭①✷ ✦❂ ✶✵✮ ④ ❝ ❂ ①✷❀ ①✸ ❂ ①✷ ✰ ✶❀ ⑥ ♣r✐♥t✭①✷✮❀
→ ❜✉t ♥❡❡❞ t♦ s♦❧✈❡ ❛ ❝♦♥str❛✐♥t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ❧♦♦♣s ❞❡s✐r❡❞ → ❡❛❝❤ ❧♦♦♣ t✉r♥ = ⇒ ♥❡✇ r❡❝✉rs✐✈❡ ❝♦♥str❛✐♥ts
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
P❛t❤ ❇❛s❡❞ ❚❉ ❣❡♥❡r❛t✐♦♥
✶ ❙❡❧❡❝t ✭❛♥♦t❤❡r✮ ♣❛t❤ Π ♦❢ t❤❡ ❈❋● ✷ ❇✉✐❧❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r❡❞✐❝❛t❡ ϕΠ ✸ ❙♦❧✈❡ ϕΠ ✭✐❢ ♣♦ss✐❜❧❡✮❀ ❦❡❡♣ ❛♥ ✐♥♣✉t s♦❧✉t✐♦♥ ❛s ❛ ❚❉ ✭✐❢ ❛♥②✮ ✹ ❇❛❝❦ t♦ ✭✶✮
❆ str❛t❡❣② ❢♦r t❤❡ ❝♦✈❡r❛❣❡ ❝r✐t❡r✐♦♥ ❆❧❧ ♣❛t❤s ✿ ❚❤❡ ❈❋● ✐s ✉♥✇♦✉♥❞ ♣r♦✈✐❞✐♥❣ ❛♥ ❡①❡❝✉t✐♦♥ tr❡❡ ❚❤❡ ❡①❡❝✉t✐♦♥ tr❡❡ ✐s ❡①♣❧♦r❡❞ ✉s✐♥❣ ❛ ❉❋❙ ❛♣♣r♦❛❝❤
✭✉♥✇✐♥❞✐♥❣✱ ✉♥❢❡❛s✐❜❧❡ ♣❛t❤s✱ ✳✳✳✮✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
P❛t❤ ❇❛s❡❞ ❚❉ ❣❡♥❡r❛t✐♦♥
✶ ❙❡❧❡❝t ✭❛♥♦t❤❡r✮ ♣❛t❤ Π ♦❢ t❤❡ ❈❋● ✷ ❇✉✐❧❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r❡❞✐❝❛t❡ ϕΠ ✸ ❙♦❧✈❡ ϕΠ ✭✐❢ ♣♦ss✐❜❧❡✮❀ ❦❡❡♣ ❛♥ ✐♥♣✉t s♦❧✉t✐♦♥ ❛s ❛ ❚❉ ✭✐❢ ❛♥②✮ ✹ ❇❛❝❦ t♦ ✭✶✮
❆ str❛t❡❣② ❢♦r t❤❡ ❝♦✈❡r❛❣❡ ❝r✐t❡r✐♦♥ ❆❧❧ ♣❛t❤s ✿ ❚❤❡ ❈❋● ✐s ✉♥✇♦✉♥❞ ♣r♦✈✐❞✐♥❣ ❛♥ ❡①❡❝✉t✐♦♥ tr❡❡ ❚❤❡ ❡①❡❝✉t✐♦♥ tr❡❡ ✐s ❡①♣❧♦r❡❞ ✉s✐♥❣ ❛ ❉❋❙ ❛♣♣r♦❛❝❤
✭✉♥✇✐♥❞✐♥❣✱ ✉♥❢❡❛s✐❜❧❡ ♣❛t❤s✱ ✳✳✳✮✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
P❛t❤ ❇❛s❡❞ ❚❉ ❣❡♥❡r❛t✐♦♥
✶ ❙❡❧❡❝t ✭❛♥♦t❤❡r✮ ♣❛t❤ Π ♦❢ t❤❡ ❈❋● ✷ ❇✉✐❧❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r❡❞✐❝❛t❡ ϕΠ ✸ ❙♦❧✈❡ ϕΠ ✭✐❢ ♣♦ss✐❜❧❡✮❀ ❦❡❡♣ ❛♥ ✐♥♣✉t s♦❧✉t✐♦♥ ❛s ❛ ❚❉ ✭✐❢ ❛♥②✮ ✹ ❇❛❝❦ t♦ ✭✶✮
❆ str❛t❡❣② ❢♦r t❤❡ ❝♦✈❡r❛❣❡ ❝r✐t❡r✐♦♥ ❆❧❧ ♣❛t❤s ✿ ❚❤❡ ❈❋● ✐s ✉♥✇♦✉♥❞ ♣r♦✈✐❞✐♥❣ ❛♥ ❡①❡❝✉t✐♦♥ tr❡❡ ❚❤❡ ❡①❡❝✉t✐♦♥ tr❡❡ ✐s ❡①♣❧♦r❡❞ ✉s✐♥❣ ❛ ❉❋❙ ❛♣♣r♦❛❝❤
✭✉♥✇✐♥❞✐♥❣✱ ✉♥❢❡❛s✐❜❧❡ ♣❛t❤s✱ ✳✳✳✮✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■❞❡❛ ✿ ❛❝❝❡❧❡r❛t❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❜② ✉s✐♥❣ ❝♦♥❝r❡t❡ ❡①❡❝✉t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡✳ → P❡r♠✐ts t♦ s❡❧❡❝t ❢❡❛s✐❜❧❡ ♣❛t❤s✳
①❁✷ ①❃❂✷ ①❁✺ ①❃❂✺ ①❁✶ ①❃❂✶
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■❞❡❛ ✿ ❛❝❝❡❧❡r❛t❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❜② ✉s✐♥❣ ❝♦♥❝r❡t❡ ❡①❡❝✉t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡✳ → P❡r♠✐ts t♦ s❡❧❡❝t ❢❡❛s✐❜❧❡ ♣❛t❤s✳
①❁✷ ①❃❂✷ ①❁✺ ①❃❂✺ ①❁✶ ①❃❂✶
❘❛♥❞♦♠ ❝❤♦✐❝❡ ✿ ❡✳❣✳ ①❂✸ ✭❝♦♥❝r❡t❡✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■❞❡❛ ✿ ❛❝❝❡❧❡r❛t❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❜② ✉s✐♥❣ ❝♦♥❝r❡t❡ ❡①❡❝✉t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡✳ → P❡r♠✐ts t♦ s❡❧❡❝t ❢❡❛s✐❜❧❡ ♣❛t❤s✳
①❁✷ ①❃❂✷ ①❁✺ ①❃❂✺ ①❁✶ ①❃❂✶
❇❛❝❦tr❛❝❦ ✰ r❡s♦❧✉t✐♦♥ ✿ x ≥ 2 ∧ x ≥ 5❀ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥ ✿ x = 8
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■❞❡❛ ✿ ❛❝❝❡❧❡r❛t❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❜② ✉s✐♥❣ ❝♦♥❝r❡t❡ ❡①❡❝✉t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡✳ → P❡r♠✐ts t♦ s❡❧❡❝t ❢❡❛s✐❜❧❡ ♣❛t❤s✳
①❁✷ ①❃❂✷ ①❁✺ ①❃❂✺ ①❁✶ ①❃❂✶ ①❂✽ ✭❝♦♥❝r❡t❡✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■❞❡❛ ✿ ❛❝❝❡❧❡r❛t❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❜② ✉s✐♥❣ ❝♦♥❝r❡t❡ ❡①❡❝✉t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡✳ → P❡r♠✐ts t♦ s❡❧❡❝t ❢❡❛s✐❜❧❡ ♣❛t❤s✳
①❁✷ ①❃❂✷ ①❁✺ ①❃❂✺ ①❁✶ \ ①❃❂✶
❇❛❝❦tr❛❝❦ ✰ r❡s♦❧✉t✐♦♥ ✿ x ≥ 2 ∧ x ≥ 5 ∧ x < 1❀ ✉♥❢❡❛s✐❜❧❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■❞❡❛ ✿ ❛❝❝❡❧❡r❛t❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❜② ✉s✐♥❣ ❝♦♥❝r❡t❡ ❡①❡❝✉t✐♦♥ ❛t t❤❡ s❛♠❡ t✐♠❡✳ → P❡r♠✐ts t♦ s❡❧❡❝t ❢❡❛s✐❜❧❡ ♣❛t❤s✳
①❁✷ ①❃❂✷ ①❁✺ ①❃❂✺ ①❁✶ ①❃❂✶
❇❛❝❦tr❛❝❦ ✰ r❡s♦❧✉t✐♦♥ ✿ x < 2❀ ❡t❝ ✳✳✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❈ ✲ ❈✰✰ ✿
❈✉t❡ ✭❯♥✐✈❡rs✐t② ♦❢ ■❧❧✐♥♦✐s✱ ❇❡r❦❡❧❡②✮ ❈r❡st ✭❇❡r❦❡❧❡②✮ ❉❛rt ✭❇❡❧❧ ▲❛❜s✮ ❊❳❊ ✭❯♥✐✈❡rs✐t② ♦❢ ❙t❛♥❢♦r❞✮ ■♥❦❛ ✭■◆❘■❆✱ ❋r❛♥❝❡✮ P❛t❤❈r❛✇❧❡r ✭❈❊❆✮
❏❛✈❛✱ ❈★ ✿
❈❆❚● ✭◆❚❚ ▲❛❜s✱ ❇❡r❦❡❧❡②✮ ❈P❇P❱ ✭■✸❙✱ ❙♦♣❤✐❛ ❆♥t✐♣♦❧✐s✱ ❋r❛♥❝❡✮ ❏❈✉t❡ ✭❯♥✐✈❡rs✐t② ♦❢ ■❧❧✐♥♦✐s✱ ❇❡r❦❡❧❡②✮ ❏❛✈❛ P❛t❤ ❋✐♥❞❡r ✭◆❆❙❆✮ P❡① ✭▼✐❝r♦s♦❢t✮ P❡t ✭❯♥✐✈❡rs✐t② ♦❢ ▼❛❞r✐❞✮
❇✐♥❛r✐❡s ✿
❖s♠♦s❡ ✭❈❊❆✮ ❙❛❣❡ ✭▼✐❝r♦s♦❢t✮ ❚r✐t♦♥ ✭❇♦r❞❡❛✉① ❯♥✐✈❡rs✐t②✱ ◗✉❛r❦s❧❛❜✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✽✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✶ ●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ✷ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ✸ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣
✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡s ✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▲❛❜❡❧❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s
✹ ❈♦♥❝❧✉s✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❙❯❚ ❊①❛♠♣❧❡ ♦❢ ❦♥♦✇♥ ♠❡t❤♦❞s ✿ ❊q✉✐✈❛❧❡♥t ❝❧❛ss❡s ❛♥❛❧②s✐s ✲ ❇♦✉♥❞❛r② ✈❛❧✉❡s ❛♥❛❧②s✐s
❉✐✈✐❞❡ t❤❡ ❣❧♦❜❛❧ ✭♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧✮ s❡t ♦❢ ✐♥♣✉ts ✐♥t♦ ❡q✉✐✈❛❧❡♥t ❝❧❛ss❡s ❖♥❡ ✈❛❧✉❡ ♦❢ t❤❡ ❝❧❛ss t❡st❡❞ = ⇒ ❛❧❧ ✈❛❧✉❡s ♦❢ t❤❡ ❝❧❛ss t❡st❡❞ ❙♦♠❡t✐♠❡s ❛❞❞ t❡sts ❢♦r t❤❡ ❜♦✉♥❞❛r✐❡s✱ ♦❢t❡♥ s♦✉r❝❡ ♦❢ ❜✉❣s ❉❡❝r❡❛s❡s t❤❡ ♥✉♠❜❡r ♦❢ ❚❉ ✐♥ t❤❡♦r②✱ s♦♠❡t✐♠❡s ♥♦t ❡❛s② ✐♥ ♣r❛❝t✐❝❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❙❯❚ ❊①❛♠♣❧❡ ♦❢ ❦♥♦✇♥ ♠❡t❤♦❞s ✿ ❈♦♠❜✐♥❛t♦r② t❡st✐♥❣ ✲ P❛✐r✇✐s❡ t❡st✐♥❣ ❲❤❡♥ ♠♦r❡ t❤❛♥ ✷ ♣❛r❛♠s✱ ❚❉ ❝❤❡❝❦s ♦♥❧② ♣❛✐rs ♦❢ ✈❛❧✉❡s✱ ♥♦t ❛❧❧ ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s❛
❊①❛♠♣❧❡ ✿ ✸ ❜♦♦❧❡❛♥ ✈❛r✐❛❜❧❡s ✿ ❱✶ ❱✷ ❱✸ ✵ ✵ ✵ ✵ ✵ ✶ ← r❡❞✉♥❞❛♥t✱ r❡♠♦✈❡ t❡st ❝❛s❡ ✵ ✶ ✵ ← r❡❞✉♥❞❛♥t✱ r❡♠♦✈❡ t❡st ❝❛s❡ ✳✳✳
❛✇✇✇✳♣❛✐r✇✐s❡✳♦r❣ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❙❯❚ ❊①❛♠♣❧❡ ♦❢ ❦♥♦✇♥ ♠❡t❤♦❞s ✿ ❘❛♥❞♦♠ t❡st✐♥❣
◗✉✐❝❦ ❢❡❡❞❜❛❝❦ ❢♦r ❝♦❛rs❡ t❡st✐♥❣ ■♥ ❝❛s❡ ♦❢ ❜✉❣ ♦♥ ❢❡✇ ✈❛❧✉❡s✱ ❧♦✇ ♣r♦❜❛❜✐❧✐t② t♦ ✜♥❞ ✐t ▼❛② ♣r♦✈✐❞❡ ❛♥ ✐♠♣♦rt❛♥t ♥✉♠❜❡r ♦❢ t❡st ❝❛s❡s → ♦r❛❝❧❡❄
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ❚❉ ✐s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❙❯❚ ❊①❛♠♣❧❡ ♦❢ ❦♥♦✇♥ ♠❡t❤♦❞s ✿ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣
P♦✇❡r❢✉❧ t❡❝❤♥✐q✉❡ P❛rt✐❝✉❧❛r❧② ❛❞❛♣t❡❞ ❢♦r t❡st✐♥❣ r❡❛❝t✐✈❡ s②st❡♠s✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♣r♦t♦❝♦❧s ◆♦t ❡❛s② t♦ ❤❛✈❡ ❛ ✭❢♦r♠❛❧✮ ♠♦❞❡❧ ✐♥ ♣r❛❝t✐❝❡ ← r❡q✉✐r❡s ❛♥ ✐♠♣♦rt❛♥t ♠♦❞❡❧❧✐♥❣ ❡✛♦rt ✭❝♦st❧②✱ ❜✉t ❣❡♥❡r❛❧❧② ♣r♦✜t❛❜❧❡✮ ❘❡q✉✐r❡s s♦♠❡t✐♠❡s ❛ ♠❛♣♣✐♥❣ ❜❡t✇❡❡♥ ❛❜str❛❝t t❡st ❝❛s❡s ❛♥❞ ❝♦♥❝r❡t❡ t❡st ❝❛s❡s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ✭▼❇❚✮ → t❡st✐♥❣ ✇✐t❤ t❤❡ ❛❜✐❧✐t② t♦ ❞❡t❡❝t ❢❛✉❧ts ✇❤✐❝❤ ❞♦ ♥♦t ❝♦♥❢♦r♠ t♦ ❛ ♠♦❞❡❧ ❝❛❧❧❡❞ s♣❡❝✐✜❝❛t✐♦♥✳
Specification conforms specifies Implementation Under Test (IUT)
→ ♣♦ss✐❜❧❡ ❛✉t♦♠❛t✐♦♥ ❢♦r t❡st ❣❡♥❡r❛t✐♦♥✱ t❡st ❡①❡❝✉t✐♦♥✱ t❡st ❡✈❛❧✉❛t✐♦♥ ✭✈❡r❞✐❝t✮ → ❋♦r♠❛❧ ▼❡t❤♦❞s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❡st ❝❛s❡s ❛r❡ ❣❡♥❡r❛t❡❞ ❢r♦♠ t❤❡ ▼♦❞❡❧ ❆s ✉s✉❛❧✱ ❚❉ ❛r❡ ❛♣♣❧✐❡❞ ♦♥ t❤❡ ■♠♣❧❡♠❡♥t❛t✐♦♥✱ ❛♥❞ r❡s✉❧ts ❛r❡ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ Pr♦❜❧❡♠s ✿
◆❡❡❞ t♦ ✜♥❞ ❛ ✏❣♦♦❞✑ ♠♦❞❡❧ ♦❢ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ❲❤❛t ❞♦❡s s♣❡❝✐❢② ♠❡❛♥❄ ❲❤❛t ❞♦❡s ❝♦♥❢♦r♠ ♠❡❛♥❄
■♠♣❧❡♠❡♥t❛t✐♦♥ ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❡q✉✐✈❛❧❡♥t t♦ ❛ ❢♦r♠❛❧ ♠♦❞❡❧ ✭❜✉t ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐s ✉♥❦♥♦✇♥✮ ◆❡❡❞ ❛ ❝♦♥❢♦r♠❛♥❝❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❙♣❡❝✐✜❝❛t✐♦♥ ❛♥❞ t❤❡ ■♠♣❧❡♠❡♥t❛t✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❡st ❝❛s❡s ❛r❡ ❣❡♥❡r❛t❡❞ ❢r♦♠ t❤❡ ▼♦❞❡❧ ❆s ✉s✉❛❧✱ ❚❉ ❛r❡ ❛♣♣❧✐❡❞ ♦♥ t❤❡ ■♠♣❧❡♠❡♥t❛t✐♦♥✱ ❛♥❞ r❡s✉❧ts ❛r❡ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ Pr♦❜❧❡♠s ✿
◆❡❡❞ t♦ ✜♥❞ ❛ ✏❣♦♦❞✑ ♠♦❞❡❧ ♦❢ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ❲❤❛t ❞♦❡s s♣❡❝✐❢② ♠❡❛♥❄ ❲❤❛t ❞♦❡s ❝♦♥❢♦r♠ ♠❡❛♥❄
■♠♣❧❡♠❡♥t❛t✐♦♥ ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❡q✉✐✈❛❧❡♥t t♦ ❛ ❢♦r♠❛❧ ♠♦❞❡❧ ✭❜✉t ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐s ✉♥❦♥♦✇♥✮ ◆❡❡❞ ❛ ❝♦♥❢♦r♠❛♥❝❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❙♣❡❝✐✜❝❛t✐♦♥ ❛♥❞ t❤❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❚✇♦ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ♦❢ r❡❛❝t✐✈❡ s②st❡♠s ✿ ❋✐♥✐t❡ ❙t❛t❡ ▼❛❝❤✐♥❡s ▲❛❜❡❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
S P ? Property P Specification S Implementation I I conf S ?
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
S P ? Property P Specification S Implementation I I conf S ? VERIFICATION
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
S P ? Property P Specification S Implementation I I conf S ? TEST
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
S P ? Property P Specification S Implementation I I conf S ? Test cases control
? ! Verdict
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
S P ? Property P Specification S Implementation I I conf S ? Test cases control
? ! Verdict Test Generation
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
S P ? Property P Specification S Implementation I I conf S ? Test cases control
? ! Verdict Test Generation Test Purpose
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋✐♥❞ ❛ ✏❣♦♦❞✑ ♠♦❞❡❧ ◆❡❝❡ss✐t② t♦ ✜♥❞ ❛ ❢♦r♠❛❧ ♠♦❞❡❧ ❛❞❛♣t❡❞ t♦ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ❚♦♦ ❛❜str❛❝t = ⇒ ♥♦t r❡❛❧✐st✐❝✱ ♥♦ ✐♥t❡r❡st ❚♦♦ ❞❡t❛✐❧❡❞ = ⇒ ❞✐✣❝✉❧t t♦ ♠♦❞❡❧✱ t♦♦ ♠❛♥② ❝❛s❡s t♦ ❝❤❡❝❦ ❯s✉❛❧❧②✱ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ♠♦❞❡❧ ❤❛s ❛♥ ✐♠♣❛❝t ♦♥ ❝❤♦✐❝❡ ♦❢ t❤❡ t❡st✐♥❣ ♠❡t❤♦❞ ✭❛♥❞ ❝♦♥✈❡rs❡❧②✮❄ ▼❛♥② ♣♦ss✐❜✐❧✐t✐❡s ♦❢ ♠♦❞❡❧s✱ ♠♦r❡ ♦r ❧❡ss ❢♦r♠❛❧
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡ ♦❢ ▼♦❞❡❧ ✿ ▼❡❛❧② ♠❛❝❤✐♥❡ ✭❋❙▼ ✇✐t❤ ♦✉t♣✉ts✮ ❙✐♠♣❧❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡ ❝♦♥tr♦❧❧❡r ❣✐✈✐♥❣ ❝❤❛♥❣❡✱ ♠❛♥❛❣✐♥❣ ✷ ❝♦✐♥s ■♥♣✉ts ✿ { ❝♦✛❡❡❘❡q❄✱ ✷✺❄✱ ✺✵❄ } ✭❧❛❜❡❧❧❡❞ ✇✐t❤ ✏❄✑✮ ❖✉t♣✉ts ✿ { ♥♦♣✦✱ ✷✺✦✱ ✺✵✦✱ ❣✐✈❡❈♦✛❡❡✦ } ✭❧❛❜❡❧❧❡❞ ✇✐t❤ ✏✦✑✮
✵ ✷✺ ✺✵ ✷✺❄ ✴ ♥♦♣✦ ✺✵❄ ✴ ♥♦♣✦ ❝♦✛❡❡❘❡q❄ ✴ ♥♦♣✦ ✷✺❄ ✴ ♥♦♣✦ ✺✵❄ ✴ ✷✺✦ ❝♦✛❡❡❘❡q❄ ✴ ♥♦♣✦ ❝♦✛❡❡❘❡q❄ ✴ ❣✐✈❡❈♦✛❡❡✦ ✷✺❄ ✴ ✷✺✦ ✺✵❄ ✴ ✺✵✦
P♦ss✐❜✐❧✐t② t♦ ❛❞❞ t✐♠❡ ✭❚❋❙▼✮✱ ❞❛t❛ ✈❛r✐❛❜❧❡s ✭❊❋❙▼✮✱ ♦r ❜♦t❤ ✭❚❊❋❙▼✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡ ♦❢ ▼♦❞❡❧ ✿ ▼❡❛❧② ♠❛❝❤✐♥❡ ✭❋❙▼ ✇✐t❤ ♦✉t♣✉ts✮ ❙✐♠♣❧❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡ ❝♦♥tr♦❧❧❡r ❣✐✈✐♥❣ ❝❤❛♥❣❡✱ ♠❛♥❛❣✐♥❣ ✷ ❝♦✐♥s ■♥♣✉ts ✿ { ❝♦✛❡❡❘❡q❄✱ ✷✺❄✱ ✺✵❄ } ✭❧❛❜❡❧❧❡❞ ✇✐t❤ ✏❄✑✮ ❖✉t♣✉ts ✿ { ♥♦♣✦✱ ✷✺✦✱ ✺✵✦✱ ❣✐✈❡❈♦✛❡❡✦ } ✭❧❛❜❡❧❧❡❞ ✇✐t❤ ✏✦✑✮
✵ ✷✺ ✺✵ ✷✺❄ ✴ ♥♦♣✦ ✺✵❄ ✴ ♥♦♣✦ ❝♦✛❡❡❘❡q❄ ✴ ♥♦♣✦ ✷✺❄ ✴ ♥♦♣✦ ✺✵❄ ✴ ✷✺✦ ❝♦✛❡❡❘❡q❄ ✴ ♥♦♣✦ ❝♦✛❡❡❘❡q❄ ✴ ❣✐✈❡❈♦✛❡❡✦ ✷✺❄ ✴ ✷✺✦ ✺✵❄ ✴ ✺✵✦
P♦ss✐❜✐❧✐t② t♦ ❛❞❞ t✐♠❡ ✭❚❋❙▼✮✱ ❞❛t❛ ✈❛r✐❛❜❧❡s ✭❊❋❙▼✮✱ ♦r ❜♦t❤ ✭❚❊❋❙▼✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡ ♦❢ ▼♦❞❡❧ ✿ ▼❡❛❧② ♠❛❝❤✐♥❡ ✭❋❙▼ ✇✐t❤ ♦✉t♣✉ts✮ ❙✐♠♣❧❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡ ❝♦♥tr♦❧❧❡r ❣✐✈✐♥❣ ❝❤❛♥❣❡✱ ♠❛♥❛❣✐♥❣ ✷ ❝♦✐♥s ■♥♣✉ts ✿ { ❝♦✛❡❡❘❡q❄✱ ✷✺❄✱ ✺✵❄ } ✭❧❛❜❡❧❧❡❞ ✇✐t❤ ✏❄✑✮ ❖✉t♣✉ts ✿ { ♥♦♣✦✱ ✷✺✦✱ ✺✵✦✱ ❣✐✈❡❈♦✛❡❡✦ } ✭❧❛❜❡❧❧❡❞ ✇✐t❤ ✏✦✑✮
✵ ✷✺ ✺✵ ✷✺❄ ✴ ♥♦♣✦ ✺✵❄ ✴ ♥♦♣✦ ❝♦✛❡❡❘❡q❄ ✴ ♥♦♣✦ ✷✺❄ ✴ ♥♦♣✦ ✺✵❄ ✴ ✷✺✦ ❝♦✛❡❡❘❡q❄ ✴ ♥♦♣✦ ❝♦✛❡❡❘❡q❄ ✴ ❣✐✈❡❈♦✛❡❡✦ ✷✺❄ ✴ ✷✺✦ ✺✵❄ ✴ ✺✵✦
P♦ss✐❜✐❧✐t② t♦ ❛❞❞ t✐♠❡ ✭❚❋❙▼✮✱ ❞❛t❛ ✈❛r✐❛❜❧❡s ✭❊❋❙▼✮✱ ♦r ❜♦t❤ ✭❚❊❋❙▼✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡ ♦❢ ▼♦❞❡❧ ✿ ▼❡❛❧② ♠❛❝❤✐♥❡ ✭❋❙▼ ✇✐t❤ ♦✉t♣✉ts✮ ❙✐♠♣❧❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡ ❝♦♥tr♦❧❧❡r ❣✐✈✐♥❣ ❝❤❛♥❣❡✱ ♠❛♥❛❣✐♥❣ ✷ ❝♦✐♥s ■♥♣✉ts ✿ { ❝♦✛❡❡❘❡q❄✱ ✷✺❄✱ ✺✵❄ } ✭❧❛❜❡❧❧❡❞ ✇✐t❤ ✏❄✑✮ ❖✉t♣✉ts ✿ { ♥♦♣✦✱ ✷✺✦✱ ✺✵✦✱ ❣✐✈❡❈♦✛❡❡✦ } ✭❧❛❜❡❧❧❡❞ ✇✐t❤ ✏✦✑✮
✵ ✷✺ ✺✵ ✷✺❄ ✴ ♥♦♣✦ ✺✵❄ ✴ ♥♦♣✦ ❝♦✛❡❡❘❡q❄ ✴ ♥♦♣✦ ✷✺❄ ✴ ♥♦♣✦ ✺✵❄ ✴ ✷✺✦ ❝♦✛❡❡❘❡q❄ ✴ ♥♦♣✦ ❝♦✛❡❡❘❡q❄ ✴ ❣✐✈❡❈♦✛❡❡✦ ✷✺❄ ✴ ✷✺✦ ✺✵❄ ✴ ✺✵✦
P♦ss✐❜✐❧✐t② t♦ ❛❞❞ t✐♠❡ ✭❚❋❙▼✮✱ ❞❛t❛ ✈❛r✐❛❜❧❡s ✭❊❋❙▼✮✱ ♦r ❜♦t❤ ✭❚❊❋❙▼✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡ ♦❢ ▼♦❞❡❧ ✿ ✭■❖✮▲❚❙ ✭▲❚❙ ✇✐t❤ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✮ ❆ ✭✈❡r② ✈❡r②✮ s✐♠♣❧✐✜❡❞ ❞✐❣✐❝♦❞❡✳ ✐♥♣✉ts ✭r❡s♣✳ ♦✉t♣✉ts✮ ❧❛❜❡❧❧❡❞ ✇✐t❤ ✏❄✑ ✭r❡s♣✳ ✏✦✑✮
s0 s1 s2 s3 s4 s5 s6 s7 ?digit ?reset τ1 !beep ?digit ?reset !open !alarm τ2 τ3 τ4
P♦ss✐❜✐❧✐t② t♦ ❛❞❞ t✐♠❡ ✭❚■❖❚❙✱ ❚❆■❖✮✱ ❞❛t❛ ✈❛r✐❛❜❧❡s ✭■❖❙❚❙✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡ ♦❢ ▼♦❞❡❧ ✿ ✭■❖✮▲❚❙ ✭▲❚❙ ✇✐t❤ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✮ ❆ ✭✈❡r② ✈❡r②✮ s✐♠♣❧✐✜❡❞ ❞✐❣✐❝♦❞❡✳ ✐♥♣✉ts ✭r❡s♣✳ ♦✉t♣✉ts✮ ❧❛❜❡❧❧❡❞ ✇✐t❤ ✏❄✑ ✭r❡s♣✳ ✏✦✑✮
s0 s1 s2 s3 s4 s5 s6 s7 ?digit ?reset τ1 !beep ?digit ?reset !open !alarm τ2 τ3 τ4
P♦ss✐❜✐❧✐t② t♦ ❛❞❞ t✐♠❡ ✭❚■❖❚❙✱ ❚❆■❖✮✱ ❞❛t❛ ✈❛r✐❛❜❧❡s ✭■❖❙❚❙✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✶ ●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ✷ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ✸ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣
✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡s ✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▲❛❜❡❧❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s
✹ ❈♦♥❝❧✉s✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
P❛rt ❡ss❡♥t✐❛❧❧② ❜❛s❡❞ ♦♥ ✿ ❬❇❏❑✰✵✺❪ ❇r♦②✱ ▼✳❀ ❏♦♥ss♦♥✱ ❇✳❀ ❑❛t♦❡♥✱ ❏✳✲P✳❀ ▲❡✉❝❦❡r✱ ▼✳✱ Pr❡ts❝❤♥❡r✱ ❆✳ ✭❊❞s✳✮✱ ✏▼♦❞❡❧✲❇❛s❡❞ ❚❡st✐♥❣ ♦❢ ❘❡❛❝t✐✈❡ ❙②st❡♠s✑✱ ❙♣r✐♥❣❡r✱ ▲◆❈❙✱ ✈♦❧✉♠❡ ✸✹✼✷ ❬❇P✾✹❪ ●✳ ✈✳ ❇♦❝❤♠❛♥♥ ❛♥❞ ❆❧❡①❛♥❞r❡ P❡tr❡♥❦♦✱ ✏Pr♦t♦❝♦❧ ❚❡st✐♥❣ ✿ ❘❡✈✐❡✇ ♦❢ ▼❡t❤♦❞s ❛♥❞ ❘❡❧❡✈❛♥❝❡ ❢♦r ❙♦❢t✇❛r❡ ❚❡st✐♥❣✑ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✶✾✾✹ ❆❈▼ ❙■●❙❖❋❚ ✐♥t❡r♥❛t✐♦♥❛❧ s②♠♣♦s✐✉♠ ♦♥ ❙♦❢t✇❛r❡ t❡st✐♥❣ ❛♥❞ ❛♥❛❧②s✐s✱ ❙❡❛tt❧❡✱ ❲❛s❤✐♥❣t♦♥✱ ❯♥✐t❡❞ ❙t❛t❡s✱ ♣ ✶✵✾ ✲ ✶✷✹✱ ✶✾✾✹ ❬❏❡r✵✹❪ ❚✳ ❏ér♦♥✱ ✏❈♦♥tr✐❜✉t✐♦♥ à ❧❛ ❣é♥ér❛t✐♦♥ ❛✉t♦♠❛t✐q✉❡ ❞❡ t❡sts ♣♦✉r ❧❡s s②stè♠❡s ré❛❝t✐❢s✱✑ ✷✵✵✹✱ ❤❛❜✐❧✐t❛t✐♦♥ à ❉✐r✐❣❡r ❞❡s ❘❡❝❤❡r❝❤❡s ✲ ❯♥✐✈❡rs✐té ❞❡ ❘❡♥♥❡s ✶✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✽✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✵ ✷✺ ✺✵ ✷✺❄✴♥♦♣✦ ✺✵❄✴♥♦♣✦ ❝❘❄✴♥♦♣✦ ✷✺❄✴♥♦♣✦ ✺✵❄✴✷✺✦ ❝❘❄✴♥♦♣✦ ❝❘❄✴❣❈✦ ✷✺❄✴✷✺✦ ✺✵❄✴✺✵✦
M = (I, O, S, δ, λ) ✇❤❡r❡ I ❛♥❞ O ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts s②♠❜♦❧s S ✐s ❛ ✜♥✐t❡ s❡t ♦❢ st❛t❡s✱ δ : S × I → S ✐s t❤❡ st❛t❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✱ ❡①t❡♥❞ t♦ ✐♥♣✉t s❡q✉❡♥❝❡s ✇✐t❤ δ∗ : S × I∗ → S∗ λ : S × I → O ✐s t❤❡ ♦✉t♣✉t ❢✉♥❝t✐♦♥✱ ❡①t❡♥❞ t♦ ♦✉t♣✉t s❡q✉❡♥❝❡s ✇✐t❤ λ∗ : S × I∗ → O∗ → ❉❡t❡r♠✐♥✐st✐❝ s✐♥❝❡ ❤❡r❡ δ ❛♥❞ λ ❛r❡ ❢✉♥❝t✐♦♥s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✵ ✷✺ ✺✵ ✷✺❄✴♥♦♣✦ ✺✵❄✴♥♦♣✦ ❝❘❄✴♥♦♣✦ ✷✺❄✴♥♦♣✦ ✺✵❄✴✷✺✦ ❝❘❄✴♥♦♣✦ ❝❘❄✴❣❈✦ ✷✺❄✴✷✺✦ ✺✵❄✴✺✵✦
M = (I, O, S, δ, λ) ✇❤❡r❡ I ❛♥❞ O ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts s②♠❜♦❧s S ✐s ❛ ✜♥✐t❡ s❡t ♦❢ st❛t❡s✱ δ : S × I → S ✐s t❤❡ st❛t❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✱ ❡①t❡♥❞ t♦ ✐♥♣✉t s❡q✉❡♥❝❡s ✇✐t❤ δ∗ : S × I∗ → S∗ λ : S × I → O ✐s t❤❡ ♦✉t♣✉t ❢✉♥❝t✐♦♥✱ ❡①t❡♥❞ t♦ ♦✉t♣✉t s❡q✉❡♥❝❡s ✇✐t❤ λ∗ : S × I∗ → O∗ → ❯s✉❛❧❧② ❝♦♠♣❧❡t❡ ✿ δ ❛♥❞ λ ❞❡✜♥❡❞ ❢♦r ❛♥② ✐♥♣✉t
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✵ ✷✺ ✺✵ ✷✺❄✴♥♦♣✦ ✺✵❄✴♥♦♣✦ ❝❘❄✴♥♦♣✦ ✷✺❄✴♥♦♣✦ ✺✵❄✴✷✺✦ ❝❘❄✴♥♦♣✦ ❝❘❄✴❣❈✦ ✷✺❄✴✷✺✦ ✺✵❄✴✺✵✦
M = (I, O, S, δ, λ) ✇❤❡r❡ I ❛♥❞ O ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts s②♠❜♦❧s S ✐s ❛ ✜♥✐t❡ s❡t ♦❢ st❛t❡s✱ δ : S × I → S ✐s t❤❡ st❛t❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✱ ❡①t❡♥❞ t♦ ✐♥♣✉t s❡q✉❡♥❝❡s ✇✐t❤ δ∗ : S × I∗ → S∗ λ : S × I → O ✐s t❤❡ ♦✉t♣✉t ❢✉♥❝t✐♦♥✱ ❡①t❡♥❞ t♦ ♦✉t♣✉t s❡q✉❡♥❝❡s ✇✐t❤ λ∗ : S × I∗ → O∗ → ❊q✉✐✈❛❧❡♥t st❛t❡s ✿ t✇♦ st❛t❡s s ❛♥❞ t ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ ∀x ∈ I∗, λ(s, x) = λ(t, x)
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✵ ✷✺ ✺✵ ✷✺❄✴♥♦♣✦ ✺✵❄✴♥♦♣✦ ❝❘❄✴♥♦♣✦ ✷✺❄✴♥♦♣✦ ✺✵❄✴✷✺✦ ❝❘❄✴♥♦♣✦ ❝❘❄✴❣❈✦ ✷✺❄✴✷✺✦ ✺✵❄✴✺✵✦
M = (I, O, S, δ, λ) ✇❤❡r❡ I ❛♥❞ O ❛r❡ ✜♥✐t❡ s❡ts ♦❢ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts s②♠❜♦❧s S ✐s ❛ ✜♥✐t❡ s❡t ♦❢ st❛t❡s✱ δ : S × I → S ✐s t❤❡ st❛t❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✱ ❡①t❡♥❞ t♦ ✐♥♣✉t s❡q✉❡♥❝❡s ✇✐t❤ δ∗ : S × I∗ → S∗ λ : S × I → O ✐s t❤❡ ♦✉t♣✉t ❢✉♥❝t✐♦♥✱ ❡①t❡♥❞ t♦ ♦✉t♣✉t s❡q✉❡♥❝❡s ✇✐t❤ λ∗ : S × I∗ → O∗ → ▼✐♥✐♠❛❧ ♠❛❝❤✐♥❡ ✿ ♥♦ ♣❛✐r ♦❢ ❞✐st✐♥❝t ❡q✉✐✈❛❧❡♥t st❛t❡s ✭♣♦ss✐❜❧❡ t♦ ❜✉✐❧❞ ❛ ♠✐♥✐♠❛❧ ♠❛❝❤✐♥❡ ❢r♦♠ ❛ ♥♦♥ ♠✐♥✐♠❛❧ ♦♥❡✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
Pr♦❜❧❡♠ ❞❡s❝r✐♣t✐♦♥ ❙♣❡❝✐✜❝❛t✐♦♥ MS✱ ▼❡❛❧② ♠❛❝❤✐♥❡✱ ❦♥♦✇♥ ■♠♣❧❡♠❡♥t❛t✐♦♥ MI ▼❡❛❧② ♠❛❝❤✐♥❡✱ ✉♥❦♥♦✇♥✱ ♦♥❧② ✐♥♣✉ts✴♦✉t♣✉ts ❛r❡ ♦❜s❡r✈❛❜❧❡ ❆✐♠ ✿ ❯s❡ t❡st s❡q✉❡♥❝❡s t♦ ❝❤❡❝❦ ✐❢ MI ✐s ❡q✉✐✈❛❧❡♥t ❛ t♦ MS✱ ✐✳❡✳ MI ❝♦♥❢♦r♠s t♦ MS
❛❍❡r❡ ✏❡q✉✐✈❛❧❡♥t✑ ♠❡❛♥s ✐s♦♠♦r♣❤✐❝
→ ❣❡♥❡r❛❧❧②✱ MS ❛♥❞ MI s✉♣♣♦s❡❞ t♦ ❜❡ ♠✐♥✐♠❛❧ ❛♥❞ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ✭✉s✉❛❧❧② ❡①✐st❡♥❝❡ ♦❢ ❛ r❡s❡t ❛❝t✐♦♥✮ → MS ❛♥❞ MI ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ✭✉s✉❛❧❧② ♥♦t ♥❡❝❡ss❛r②✮
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋❛✉❧t ♠♦❞❡❧ ❈❤❡❝❦✐♥❣ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ MI ❛♥❞ MS ♠❡❛♥s ❝❤❡❝❦✐♥❣ ✐❢ MI ❤❛s ♥♦ ✿ ❖✉t♣✉t ❢❛✉❧t ✿ ♥♦t t❤❡ ❡①♣❡❝t❡❞ ♦✉t♣✉t ❢♦r ❛ ❣✐✈❡♥ ✭st❛t❡✱ ✐♥♣✉t✮ ❚r❛♥s❢❡r ❢❛✉❧t ✿ ♥♦t t❤❡ ❡①♣❡❝t❡❞ ❛rr✐✈❛❧ st❛t❡ ❢♦r ❛ ❣✐✈❡♥ ✭st❛t❡✱ ✐♥♣✉t✮ ❊①❤❛✉st✐✈✐t② = ⇒ ❛♥② tr❛♥s✐t✐♦♥ s❤♦✉❧❞ ❜❡ ❝❤❡❝❦❡❞ ❊❧❡♠❡♥t❛r② t❡st✱ ❣❡♥❡r❛❧ ❛❧❣♦r✐t❤♠ ❋♦r ❛♥② st❛t❡ s ❛♥❞ ❛♥② ✐♥♣✉t i ✭♦❢ t❤❡ s♣❡❝✐✜❝❛t✐♦♥✮
❆♣♣❧② i✱ ✈❡r✐❢② ♦✉t♣✉t o ✭❝♦♠♣❛r❡ t♦ t❤❡ s♣❡❝✐✜❝❛t✐♦♥✮ ■❞❡♥t✐❢② ❛rr✐✈❛❧ st❛t❡ ✇✐t❤ ❛ s❡q✉❡♥❝❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋❛✉❧t ♠♦❞❡❧ ❈❤❡❝❦✐♥❣ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ MI ❛♥❞ MS ♠❡❛♥s ❝❤❡❝❦✐♥❣ ✐❢ MI ❤❛s ♥♦ ✿ ❖✉t♣✉t ❢❛✉❧t ✿ ♥♦t t❤❡ ❡①♣❡❝t❡❞ ♦✉t♣✉t ❢♦r ❛ ❣✐✈❡♥ ✭st❛t❡✱ ✐♥♣✉t✮ ❚r❛♥s❢❡r ❢❛✉❧t ✿ ♥♦t t❤❡ ❡①♣❡❝t❡❞ ❛rr✐✈❛❧ st❛t❡ ❢♦r ❛ ❣✐✈❡♥ ✭st❛t❡✱ ✐♥♣✉t✮ ❊①❤❛✉st✐✈✐t② = ⇒ ❛♥② tr❛♥s✐t✐♦♥ s❤♦✉❧❞ ❜❡ ❝❤❡❝❦❡❞ ❊❧❡♠❡♥t❛r② t❡st✱ ❣❡♥❡r❛❧ ❛❧❣♦r✐t❤♠ ❋♦r ❛♥② st❛t❡ s ❛♥❞ ❛♥② ✐♥♣✉t i ✭♦❢ t❤❡ s♣❡❝✐✜❝❛t✐♦♥✮
❆♣♣❧② i✱ ✈❡r✐❢② ♦✉t♣✉t o ✭❝♦♠♣❛r❡ t♦ t❤❡ s♣❡❝✐✜❝❛t✐♦♥✮ ■❞❡♥t✐❢② ❛rr✐✈❛❧ st❛t❡ ✇✐t❤ ❛ s❡q✉❡♥❝❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋♦r ❛ ❣✐✈❡♥ ▼❡❛❧② ▼❛❝❤✐♥❡ M = (I, O, S, δ, λ)✱ ❉✐st✐♥❣✉✐s❤✐♥❣ ❙❡q✉❡♥❝❡ ✿ ∃DS ∈ I∗, ∀s, s′ ∈ S : s = s′ ⇒ λ∗(s, DS) = λ∗(s′, DS) ✭❉❙ ♠❡t❤♦❞ ❬●♦♥✼✵❪✮ ❯■❖✷ ❙❡q✉❡♥❝❡ ✿ ∀s ∈ S, ∃UIOs ∈ I∗, ∀s′ ∈ S \ {s}, λ∗(s, UIOs) = λ∗(s′, UIOs) ✭❯■❖ ♠❡t❤♦❞ ❬❙❉✽✽❪✮ ❲ s❡t ♦❢ xij s❡q✉❡♥❝❡s ✿ ∀si, sj ∈ S, ∃xij ∈ I∗, λ∗(si, xij) = λ∗(sj, xij) ✭❲ ♠❡t❤♦❞ ❬❈❤♦✼✽❪✮ → ❲ s❡t ❛❧✇❛②s ❡①✐sts ✐♥ ❝❛s❡ ♦❢ ♠✐♥✐♠❛❧ ♠❛❝❤✐♥❡✱ ♦t❤❡rs ♠❛② ♥♦t ❡①✐st
✷❯♥✐q✉❡ ■♥♣✉t ❖✉t♣✉t ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋♦r ❛ ❣✐✈❡♥ ▼❡❛❧② ▼❛❝❤✐♥❡ M = (I, O, S, δ, λ)✱ ❉✐st✐♥❣✉✐s❤✐♥❣ ❙❡q✉❡♥❝❡ ✿ ∃DS ∈ I∗, ∀s, s′ ∈ S : s = s′ ⇒ λ∗(s, DS) = λ∗(s′, DS) ✭❉❙ ♠❡t❤♦❞ ❬●♦♥✼✵❪✮ ❯■❖✷ ❙❡q✉❡♥❝❡ ✿ ∀s ∈ S, ∃UIOs ∈ I∗, ∀s′ ∈ S \ {s}, λ∗(s, UIOs) = λ∗(s′, UIOs) ✭❯■❖ ♠❡t❤♦❞ ❬❙❉✽✽❪✮ ❲ s❡t ♦❢ xij s❡q✉❡♥❝❡s ✿ ∀si, sj ∈ S, ∃xij ∈ I∗, λ∗(si, xij) = λ∗(sj, xij) ✭❲ ♠❡t❤♦❞ ❬❈❤♦✼✽❪✮ → ❲ s❡t ❛❧✇❛②s ❡①✐sts ✐♥ ❝❛s❡ ♦❢ ♠✐♥✐♠❛❧ ♠❛❝❤✐♥❡✱ ♦t❤❡rs ♠❛② ♥♦t ❡①✐st
✷❯♥✐q✉❡ ■♥♣✉t ❖✉t♣✉t ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋♦r ❛ ❣✐✈❡♥ ▼❡❛❧② ▼❛❝❤✐♥❡ M = (I, O, S, δ, λ)✱ ❉✐st✐♥❣✉✐s❤✐♥❣ ❙❡q✉❡♥❝❡ ✿ ∃DS ∈ I∗, ∀s, s′ ∈ S : s = s′ ⇒ λ∗(s, DS) = λ∗(s′, DS) ✭❉❙ ♠❡t❤♦❞ ❬●♦♥✼✵❪✮ ❯■❖✷ ❙❡q✉❡♥❝❡ ✿ ∀s ∈ S, ∃UIOs ∈ I∗, ∀s′ ∈ S \ {s}, λ∗(s, UIOs) = λ∗(s′, UIOs) ✭❯■❖ ♠❡t❤♦❞ ❬❙❉✽✽❪✮ ❲ s❡t ♦❢ xij s❡q✉❡♥❝❡s ✿ ∀si, sj ∈ S, ∃xij ∈ I∗, λ∗(si, xij) = λ∗(sj, xij) ✭❲ ♠❡t❤♦❞ ❬❈❤♦✼✽❪✮ → ❲ s❡t ❛❧✇❛②s ❡①✐sts ✐♥ ❝❛s❡ ♦❢ ♠✐♥✐♠❛❧ ♠❛❝❤✐♥❡✱ ♦t❤❡rs ♠❛② ♥♦t ❡①✐st
✷❯♥✐q✉❡ ■♥♣✉t ❖✉t♣✉t ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋♦r ❛ ❣✐✈❡♥ ▼❡❛❧② ▼❛❝❤✐♥❡ M = (I, O, S, δ, λ)✱ ❉✐st✐♥❣✉✐s❤✐♥❣ ❙❡q✉❡♥❝❡ ✿ ∃DS ∈ I∗, ∀s, s′ ∈ S : s = s′ ⇒ λ∗(s, DS) = λ∗(s′, DS) ✭❉❙ ♠❡t❤♦❞ ❬●♦♥✼✵❪✮ ❯■❖✷ ❙❡q✉❡♥❝❡ ✿ ∀s ∈ S, ∃UIOs ∈ I∗, ∀s′ ∈ S \ {s}, λ∗(s, UIOs) = λ∗(s′, UIOs) ✭❯■❖ ♠❡t❤♦❞ ❬❙❉✽✽❪✮ ❲ s❡t ♦❢ xij s❡q✉❡♥❝❡s ✿ ∀si, sj ∈ S, ∃xij ∈ I∗, λ∗(si, xij) = λ∗(sj, xij) ✭❲ ♠❡t❤♦❞ ❬❈❤♦✼✽❪✮ → ❲ s❡t ❛❧✇❛②s ❡①✐sts ✐♥ ❝❛s❡ ♦❢ ♠✐♥✐♠❛❧ ♠❛❝❤✐♥❡✱ ♦t❤❡rs ♠❛② ♥♦t ❡①✐st
✷❯♥✐q✉❡ ■♥♣✉t ❖✉t♣✉t ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❆✐♠ ✿ r❡❞✉❝❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ t❡st ❝❛s❡ ❚❚ ❬◆❚✽✶❪ ❋✐♥❞ ❛ ♠✐♥✐♠❛❧ s❡q✉❡♥❝❡ r✉♥♥✐♥❣ t❤r♦✉❣❤ ❛❧❧ tr❛♥s✐t✐♦♥s → ❈❤✐♥❡s❡ P♦st♠❛♥ ♣r♦❜❧❡♠ ✿
❚r❛♥s❢♦r♠ t❤❡ ❣r❛♣❤ ✐♥t♦ ❛ s②♠♠❡tr✐❝ ♦♥❡ ✐♥ ❛ ♠✐♥✐♠❛❧ ✇❛② ❋✐♥❞ ❛♥ ❊✉❧❡r✐❛♥ ❝✐r❝✉✐t
❯■❖ ❬❙❉✽✽❪
❋✐♥❞ ❛ ❯■❖ s❡q✉❡♥❝❡ ❢♦r ❡❛❝❤ st❛t❡ ❈♦♥s✐❞❡r✐♥❣ t❤❡ tr❛♥s✐t✐♦♥s si
i/o
→ sj
UIOsj
→ sk✱ ✜♥❞ ❛ ♠✐♥✐♠❛❧ ❝✐r❝✉✐t r✉♥♥✐♥❣ ♦♥❝❡ t❤r♦✉❣❤ t❤❡s❡ tr❛♥s✐t✐♦♥s → ❘✉r❛❧ ❈❤✐♥❡s❡ P♦st♠❛♥ ♣r♦❜❧❡♠
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❆✐♠ ✿ r❡❞✉❝❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ t❡st ❝❛s❡ ❚❚ ❬◆❚✽✶❪ ❋✐♥❞ ❛ ♠✐♥✐♠❛❧ s❡q✉❡♥❝❡ r✉♥♥✐♥❣ t❤r♦✉❣❤ ❛❧❧ tr❛♥s✐t✐♦♥s → ❈❤✐♥❡s❡ P♦st♠❛♥ ♣r♦❜❧❡♠ ✿
❚r❛♥s❢♦r♠ t❤❡ ❣r❛♣❤ ✐♥t♦ ❛ s②♠♠❡tr✐❝ ♦♥❡ ✐♥ ❛ ♠✐♥✐♠❛❧ ✇❛② ❋✐♥❞ ❛♥ ❊✉❧❡r✐❛♥ ❝✐r❝✉✐t
❯■❖ ❬❙❉✽✽❪
❋✐♥❞ ❛ ❯■❖ s❡q✉❡♥❝❡ ❢♦r ❡❛❝❤ st❛t❡ ❈♦♥s✐❞❡r✐♥❣ t❤❡ tr❛♥s✐t✐♦♥s si
i/o
→ sj
UIOsj
→ sk✱ ✜♥❞ ❛ ♠✐♥✐♠❛❧ ❝✐r❝✉✐t r✉♥♥✐♥❣ ♦♥❝❡ t❤r♦✉❣❤ t❤❡s❡ tr❛♥s✐t✐♦♥s → ❘✉r❛❧ ❈❤✐♥❡s❡ P♦st♠❛♥ ♣r♦❜❧❡♠
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❉❙ ❬●♦♥✼✵❪ ❈♦✉❧❞ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❯■❖ ♠❡t❤♦❞✱ ✇✐t❤ t❤❡ s❛♠❡ ❯■❖ ❢♦r ❡❛❝❤ st❛t❡ ❲ ❬❈❤♦✼✽❪ ✸
❋✐♥❞ ❛ ❚r❛♥s✐t✐♦♥ ❈♦✈❡r ❙❡t P ✿ s❡t ♦❢ ✐♥♣✉t s❡q✉❡♥❝❡s s✳t✳ ❢♦r ❡❛❝❤ st❛t❡ s ∈ S ❛♥❞ ❡❛❝❤ ✐♥♣✉t a ∈ I✱ t❤❡r❡ ❡①✐sts ❛♥ ✐♥♣✉t s❡q✉❡♥❝❡ ✐♥ P st❛rt✐♥❣ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ st❛t❡ ❛♥❞ ❡♥❞✐♥❣ ✇✐t❤ t❤❡ tr❛♥s✐t✐♦♥ t❤❛t ❛♣♣❧✐❡s a t♦ s✳ ❋✐♥❞ ❛ ❈❤❛r❛❝t❡r✐s✐♥❣ ❙❡t W ✿ s❡t ♦❢ ✐♥♣✉t s❡q✉❡♥❝❡s s✳t✳ ∀si, sj ∈ S, ∃xij ∈ W, λ∗(si, xij) = λ∗(sj, xij) ◆♦t✐♥❣ X · Y t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ X ✇✐t❤ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ Y ✱ ❣❡♥❡r❛t❡ {reset} · P · W
Wp✱ UIOp✱ UIOv✱ DS ✇✐t❤♦✉t r❡s❡ts✱ ❆❞❛♣t❛t✐✈❡ DS✱ HSI✱ ✳✳✳ P♦ss✐❜✐❧✐t② t♦ ❜❡ ♠♦r❡ ❡✣❝✐❡♥t ❜② ✉s✐♥❣ ❆❞❛♣t❛t✐✈❡ ❙❡q✉❡♥❝❡s
✸❝❛s❡ ✇❤❡r❡ MI ❛♥❞ MS ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❉❙ ❬●♦♥✼✵❪ ❈♦✉❧❞ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❯■❖ ♠❡t❤♦❞✱ ✇✐t❤ t❤❡ s❛♠❡ ❯■❖ ❢♦r ❡❛❝❤ st❛t❡ ❲ ❬❈❤♦✼✽❪ ✸
❋✐♥❞ ❛ ❚r❛♥s✐t✐♦♥ ❈♦✈❡r ❙❡t P ✿ s❡t ♦❢ ✐♥♣✉t s❡q✉❡♥❝❡s s✳t✳ ❢♦r ❡❛❝❤ st❛t❡ s ∈ S ❛♥❞ ❡❛❝❤ ✐♥♣✉t a ∈ I✱ t❤❡r❡ ❡①✐sts ❛♥ ✐♥♣✉t s❡q✉❡♥❝❡ ✐♥ P st❛rt✐♥❣ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ st❛t❡ ❛♥❞ ❡♥❞✐♥❣ ✇✐t❤ t❤❡ tr❛♥s✐t✐♦♥ t❤❛t ❛♣♣❧✐❡s a t♦ s✳ ❋✐♥❞ ❛ ❈❤❛r❛❝t❡r✐s✐♥❣ ❙❡t W ✿ s❡t ♦❢ ✐♥♣✉t s❡q✉❡♥❝❡s s✳t✳ ∀si, sj ∈ S, ∃xij ∈ W, λ∗(si, xij) = λ∗(sj, xij) ◆♦t✐♥❣ X · Y t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ X ✇✐t❤ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ Y ✱ ❣❡♥❡r❛t❡ {reset} · P · W
Wp✱ UIOp✱ UIOv✱ DS ✇✐t❤♦✉t r❡s❡ts✱ ❆❞❛♣t❛t✐✈❡ DS✱ HSI✱ ✳✳✳ P♦ss✐❜✐❧✐t② t♦ ❜❡ ♠♦r❡ ❡✣❝✐❡♥t ❜② ✉s✐♥❣ ❆❞❛♣t❛t✐✈❡ ❙❡q✉❡♥❝❡s
✸❝❛s❡ ✇❤❡r❡ MI ❛♥❞ MS ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❉❙ ❬●♦♥✼✵❪ ❈♦✉❧❞ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❯■❖ ♠❡t❤♦❞✱ ✇✐t❤ t❤❡ s❛♠❡ ❯■❖ ❢♦r ❡❛❝❤ st❛t❡ ❲ ❬❈❤♦✼✽❪ ✸
❋✐♥❞ ❛ ❚r❛♥s✐t✐♦♥ ❈♦✈❡r ❙❡t P ✿ s❡t ♦❢ ✐♥♣✉t s❡q✉❡♥❝❡s s✳t✳ ❢♦r ❡❛❝❤ st❛t❡ s ∈ S ❛♥❞ ❡❛❝❤ ✐♥♣✉t a ∈ I✱ t❤❡r❡ ❡①✐sts ❛♥ ✐♥♣✉t s❡q✉❡♥❝❡ ✐♥ P st❛rt✐♥❣ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ st❛t❡ ❛♥❞ ❡♥❞✐♥❣ ✇✐t❤ t❤❡ tr❛♥s✐t✐♦♥ t❤❛t ❛♣♣❧✐❡s a t♦ s✳ ❋✐♥❞ ❛ ❈❤❛r❛❝t❡r✐s✐♥❣ ❙❡t W ✿ s❡t ♦❢ ✐♥♣✉t s❡q✉❡♥❝❡s s✳t✳ ∀si, sj ∈ S, ∃xij ∈ W, λ∗(si, xij) = λ∗(sj, xij) ◆♦t✐♥❣ X · Y t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ X ✇✐t❤ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ Y ✱ ❣❡♥❡r❛t❡ {reset} · P · W
Wp✱ UIOp✱ UIOv✱ DS ✇✐t❤♦✉t r❡s❡ts✱ ❆❞❛♣t❛t✐✈❡ DS✱ HSI✱ ✳✳✳ P♦ss✐❜✐❧✐t② t♦ ❜❡ ♠♦r❡ ❡✣❝✐❡♥t ❜② ✉s✐♥❣ ❆❞❛♣t❛t✐✈❡ ❙❡q✉❡♥❝❡s
✸❝❛s❡ ✇❤❡r❡ MI ❛♥❞ MS ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❉❙ ❬●♦♥✼✵❪ ❈♦✉❧❞ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❯■❖ ♠❡t❤♦❞✱ ✇✐t❤ t❤❡ s❛♠❡ ❯■❖ ❢♦r ❡❛❝❤ st❛t❡ ❲ ❬❈❤♦✼✽❪ ✸
❋✐♥❞ ❛ ❚r❛♥s✐t✐♦♥ ❈♦✈❡r ❙❡t P ✿ s❡t ♦❢ ✐♥♣✉t s❡q✉❡♥❝❡s s✳t✳ ❢♦r ❡❛❝❤ st❛t❡ s ∈ S ❛♥❞ ❡❛❝❤ ✐♥♣✉t a ∈ I✱ t❤❡r❡ ❡①✐sts ❛♥ ✐♥♣✉t s❡q✉❡♥❝❡ ✐♥ P st❛rt✐♥❣ ❢r♦♠ t❤❡ ✐♥✐t✐❛❧ st❛t❡ ❛♥❞ ❡♥❞✐♥❣ ✇✐t❤ t❤❡ tr❛♥s✐t✐♦♥ t❤❛t ❛♣♣❧✐❡s a t♦ s✳ ❋✐♥❞ ❛ ❈❤❛r❛❝t❡r✐s✐♥❣ ❙❡t W ✿ s❡t ♦❢ ✐♥♣✉t s❡q✉❡♥❝❡s s✳t✳ ∀si, sj ∈ S, ∃xij ∈ W, λ∗(si, xij) = λ∗(sj, xij) ◆♦t✐♥❣ X · Y t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ X ✇✐t❤ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ Y ✱ ❣❡♥❡r❛t❡ {reset} · P · W
Wp✱ UIOp✱ UIOv✱ DS ✇✐t❤♦✉t r❡s❡ts✱ ❆❞❛♣t❛t✐✈❡ DS✱ HSI✱ ✳✳✳ P♦ss✐❜✐❧✐t② t♦ ❜❡ ♠♦r❡ ❡✣❝✐❡♥t ❜② ✉s✐♥❣ ❆❞❛♣t❛t✐✈❡ ❙❡q✉❡♥❝❡s
✸❝❛s❡ ✇❤❡r❡ MI ❛♥❞ MS ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✶ ●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ✷ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ✸ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣
✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡s ✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▲❛❜❡❧❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s
✹ ❈♦♥❝❧✉s✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
P❛rt ❡ss❡♥t✐❛❧❧② ❜❛s❡❞ ♦♥ ✿ ❬❚r❡✾✻❪ ❏✳ ❚r❡t♠❛♥s✱ ✏❚❡st ❣❡♥❡r❛t✐♦♥ ✇✐t❤ ✐♥♣✉ts✱ ♦✉t♣✉ts✱ ❛♥❞ r❡♣❡t✐t✐✈❡ q✉✐❡s❝❡♥❝❡✱✑ ❙♦❢t✇❛r❡✕❈♦♥❝❡♣ts ❛♥❞ ❚♦♦❧s✱ ✈♦❧✳ ✶✼✱ ♣♣✳ ✶✵✸✕✶✷✵✱ ✶✾✾✻✳ ❬❏❏✵✹❪ ❈✳ ❏❛r❞ ❛♥❞ ❚✳ ❏ér♦♥✱ ✏❚❣✈✿ t❤❡♦r②✱ ♣r✐♥❝✐♣❧❡s ❛♥❞ ❛❧❣♦r✐t❤♠s✱ ❛ t♦♦❧ ❢♦r t❤❡ ❛✉t♦♠❛t✐❝ s②♥t❤❡s✐s ♦❢ ❝♦♥❢♦r♠❛♥❝❡ t❡st ❝❛s❡s ❢♦r ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ r❡❛❝t✐✈❡ s②st❡♠s✱✑ ❙♦❢t✇❛r❡ ❚♦♦❧s ❢♦r ❚❡❝❤♥♦❧♦❣② ❚r❛♥s❢❡r ✭❙❚❚❚✮✱ ✶✵ ✷✵✵✹✳ ❬❏❡r✵✹❪ ❚✳ ❏ér♦♥✱ ✏❈♦♥tr✐❜✉t✐♦♥ à ❧❛ ❣é♥ér❛t✐♦♥ ❛✉t♦♠❛t✐q✉❡ ❞❡ t❡sts ♣♦✉r ❧❡s s②stè♠❡s ré❛❝t✐❢s✱✑ ✷✵✵✹✱ ❤❛❜✐❧✐t❛t✐♦♥ à ❉✐r✐❣❡r ❞❡s ❘❡❝❤❡r❝❤❡s ✲ ❯♥✐✈❡rs✐té ❞❡ ❘❡♥♥❡s ✶✳ ❬❏❡r✶✷❪ ❚✳ ❏ér♦♥✱ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❝♦✉rs❡s✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
s0 s1 s2 s3 s4 s5 s6 s7 ?digit ?reset τ1 !beep ?digit ?reset !open !alarm τ2 τ3 τ4
M = (QM, AM, − →M, qM
0 ) ✇✐t❤ ✿
QM s❡t ♦❢ st❛t❡s qM
0 ∈ QM ✐♥✐t✐❛❧ st❛t❡
AM ❛❝t✐♦♥ ❛❧♣❤❛❜❡t✱
AM
I
✐♥♣✉t ❛❧♣❤❛❜❡t ✭✇✐t❤ ?✮ AM
O ♦✉t♣✉t ❛❧♣❤❛❜❡t ✭✇✐t❤ !✮
IM ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ✭τk✮
− →M⊆ QM × AM × QM tr❛♥s✐t✐♦♥ r❡❧❛t✐♦♥ AM
V IS = AM I ∪ AM O s❡t ♦❢ ✈✐s✐❜❧❡ ❛❝t✐♦♥s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
s0 s1 s2 s3 s4 s5 s6 s7 ?d ?r τ1 !b ?d ?r !o !a τ2 τ3 τ4
M = (Q, A, − →, q0) ✇✐t❤ ✿ Q s❡t ♦❢ st❛t❡s q0 ∈ Q ✐♥✐t✐❛❧ st❛t❡ A ❛❝t✐♦♥ ❛❧♣❤❛❜❡t✱
AI ✐♥♣✉t ❛❧♣❤❛❜❡t ✭✇✐t❤ ?✮ AO ♦✉t♣✉t ❛❧♣❤❛❜❡t ✭✇✐t❤ !✮ I ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ✭τk✮
− →⊆ Q × A × Q tr❛♥s✐t✐♦♥ r❡❧❛t✐♦♥ AV IS = AI ∪ AO s❡t ♦❢ ✈✐s✐❜❧❡ ❛❝t✐♦♥s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
s0 s1 s2 s3 s4 s5 s6 s7 ?d ?r τ1 !b ?d ?r !o !a τ2 τ3 τ4
❘✉♥s✿ ❛❧t❡r♥❛t❡ s❡q✉❡♥❝❡s ♦❢ st❛t❡s ❛♥❞ ❛❝t✐♦♥s ✜r❡❛❜❧❡ ❜t✇ t❤♦s❡ st❛t❡s s0
?d
→ s1
τ1
→ s2
?d
→ s3
!o
→ s4 ∈ Runs(M) Traces✿ ♣r♦❥❡❝t✐♦♥s ♦❢ Runs ♦♥ ✈✐s✐❜❧❡ ❛❝t✐♦♥s✿
Traces(M) = {ε, ?d, ?r, ?d.?r, ?r.?d, ?d.!b, ...}
P ❛❢t❡r σ✿ s❡t ♦❢ st❛t❡s r❡❛❝❤❛❜❧❡ ❢r♦♠ P ❛❢t❡r ♦❜s❡r✈❛t✐♦♥ σ✿ {s2} ❛❢t❡r ?d.!o = {s0, s4} {s0} ❛❢t❡r ?d, !a = ∅ M ❛❢t❡r σ {q0} ❛❢t❡r σ
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✽✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
M ✐s ❞❡t❡r♠✐♥✐st✐❝ ✐❢ ✐t ❤❛s ♥♦ ✐♥t❡r♥❛❧ ❛❝t✐♦♥✱ ❛♥❞ ∀q, q′, q′′ ∈ Q, ∀a ∈ AV IS, (q
a
− → q′ ∧ q
a
− → q′′) ⇒ q′ = q′′
s1 s2 s3 ?x !a !a s1 s2 s3 s4 τ !a !b
◆♦t t♦ ❜❡ ❝♦♥❢✉s❡❞ ✇✐t❤ ✉♥❝♦♥tr♦❧❧❡❞ ❝❤♦✐❝❡
s1 s2 s3 ?x !a !b
❉❡t❡r♠✐♥✐③❛t✐♦♥✿ det(M) = (2Q, AV IS, − →det, q0 ❛❢t❡r ǫ) ✇✐t❤ P
a
− →det P ′ ⇔ P, P ′ ∈ 2Q, a ∈ AV IS ❛♥❞ P ′ = P ❛❢t❡r a✳ Traces(M) = Traces(det(M))
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
M ✐s ❞❡t❡r♠✐♥✐st✐❝ ✐❢ ✐t ❤❛s ♥♦ ✐♥t❡r♥❛❧ ❛❝t✐♦♥✱ ❛♥❞ ∀q, q′, q′′ ∈ Q, ∀a ∈ AV IS, (q
a
− → q′ ∧ q
a
− → q′′) ⇒ q′ = q′′
s1 s2 s3 ?x !a !a s1 s2 s3 s4 τ !a !b
◆♦t t♦ ❜❡ ❝♦♥❢✉s❡❞ ✇✐t❤ ✉♥❝♦♥tr♦❧❧❡❞ ❝❤♦✐❝❡
s1 s2 s3 ?x !a !b
❉❡t❡r♠✐♥✐③❛t✐♦♥✿ det(M) = (2Q, AV IS, − →det, q0 ❛❢t❡r ǫ) ✇✐t❤ P
a
− →det P ′ ⇔ P, P ′ ∈ 2Q, a ∈ AV IS ❛♥❞ P ′ = P ❛❢t❡r a✳ Traces(M) = Traces(det(M))
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■♥ t❡st✐♥❣ ♣r❛❝t✐❝❡✱ ♦♥❡ ❝❛♥ ♦❜s❡r✈❡ tr❛❝❡s ♦❢ t❤❡ IUT✱ ❜✉t ❛❧s♦ ✐ts q✉✐❡s❝❡♥❝❡s ✇✐t❤ t✐♠❡rs✳ ❖♥❧② q✉✐❡s❝❡♥❝❡s ♦❢ IUT ✉♥s♣❡❝✐✜❡❞ ✐♥ S s❤♦✉❧❞ ❜❡ r❡❥❡❝t❡❞✳
s0 s1 s2 s3 s4 s5 s6 s7 ?d ?r τ1 !b ?d ?r !o !a τ2 τ3 τ4
◆♦t❛t✐♦♥ ✿ Γ(q) {a ∈ A | q
a
→} ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ Γ(q) = ∅✳ ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠s ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ Γ(q) ⊆ AI✳ ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ ∃τ1, ...τn : q τ1...τn − → q✳ quiescent(M) = deadlock(M) ∪ livelock(M) ∪ outputlock(M)
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■♥ t❡st✐♥❣ ♣r❛❝t✐❝❡✱ ♦♥❡ ❝❛♥ ♦❜s❡r✈❡ tr❛❝❡s ♦❢ t❤❡ IUT✱ ❜✉t ❛❧s♦ ✐ts q✉✐❡s❝❡♥❝❡s ✇✐t❤ t✐♠❡rs✳ ❖♥❧② q✉✐❡s❝❡♥❝❡s ♦❢ IUT ✉♥s♣❡❝✐✜❡❞ ✐♥ S s❤♦✉❧❞ ❜❡ r❡❥❡❝t❡❞✳
s0 s1 s2 s3 s4 s5 s6 s7 ?d ?r τ1 !b ?d ?r !o !a τ2 τ3 τ4
◆♦t❛t✐♦♥ ✿ Γ(q) {a ∈ A | q
a
→} ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ Γ(q) = ∅✳ ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠s ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ Γ(q) ⊆ AI ✳ ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ ∃τ1, ...τn : q τ1...τn − → q✳ quiescent(M) = deadlock(M) ∪ livelock(M) ∪ outputlock(M)
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■♥ t❡st✐♥❣ ♣r❛❝t✐❝❡✱ ♦♥❡ ❝❛♥ ♦❜s❡r✈❡ tr❛❝❡s ♦❢ t❤❡ IUT✱ ❜✉t ❛❧s♦ ✐ts q✉✐❡s❝❡♥❝❡s ✇✐t❤ t✐♠❡rs✳ ❖♥❧② q✉✐❡s❝❡♥❝❡s ♦❢ IUT ✉♥s♣❡❝✐✜❡❞ ✐♥ S s❤♦✉❧❞ ❜❡ r❡❥❡❝t❡❞✳
s0 s1 s2 s3 s4 s5 s6 s7 ?d ?r τ1 !b ?d ?r !o !a τ2 τ3 τ4
◆♦t❛t✐♦♥ ✿ Γ(q) {a ∈ A | q
a
→} ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ Γ(q) = ∅✳ ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠ ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ Γ(q) ⊆ AI✳ ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ ∃τ1, ...τn : q τ1...τn − → q✳ quiescent(M) = deadlock(M) ∪ livelock(M) ∪ outputlock(M)
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■♥ t❡st✐♥❣ ♣r❛❝t✐❝❡✱ ♦♥❡ ❝❛♥ ♦❜s❡r✈❡ tr❛❝❡s ♦❢ t❤❡ IUT✱ ❜✉t ❛❧s♦ ✐ts q✉✐❡s❝❡♥❝❡s ✇✐t❤ t✐♠❡rs✳ ❖♥❧② q✉✐❡s❝❡♥❝❡s ♦❢ IUT ✉♥s♣❡❝✐✜❡❞ ✐♥ S s❤♦✉❧❞ ❜❡ r❡❥❡❝t❡❞✳
s0 s1 s2 s3 s4 s5 s6 s7 ?d ?r τ1 !b ?d ?r !o !a τ2 τ3 τ4
◆♦t❛t✐♦♥ ✿ Γ(q) {a ∈ A | q
a
→} ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ Γ(q) = ∅✳ ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠ ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ Γ(q) ⊆ AI✳ ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ ∃τ1, ...τn : q τ1...τn − → q✳ quiescent(M) = deadlock(M) ∪ livelock(M) ∪ outputlock(M)
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
■♥ t❡st✐♥❣ ♣r❛❝t✐❝❡✱ ♦♥❡ ❝❛♥ ♦❜s❡r✈❡ tr❛❝❡s ♦❢ t❤❡ IUT✱ ❜✉t ❛❧s♦ ✐ts q✉✐❡s❝❡♥❝❡s ✇✐t❤ t✐♠❡rs✳ ❖♥❧② q✉✐❡s❝❡♥❝❡s ♦❢ IUT ✉♥s♣❡❝✐✜❡❞ ✐♥ S s❤♦✉❧❞ ❜❡ r❡❥❡❝t❡❞✳
s0 s1 s2 s3 s4 s5 s6 s7 ?d ?r τ1 !b ?d ?r !o !a τ2 τ3 τ4
◆♦t❛t✐♦♥ ✿ Γ(q) {a ∈ A | q
a
→} ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ Γ(q) = ∅✳ ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠ ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ Γ(q) ⊆ AI✳ ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ ∃τ1, ...τn : q τ1...τn − → q✳ quiescent(M) = deadlock(M) ∪ livelock(M) ∪ outputlock(M)
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
◗✉✐❡s❝❡♥❝❡ ✿ s♣❡❝✐❛❧ ♦✉t♣✉t δ ❙✉s♣❡♥s✐♦♥ ❛✉t♦♠❛t♦♥ ∆(M) ∆(M) ❂ ❙♣❡❝✐✜❝❛t✐♦♥ M ✰ δ✲tr❛♥s✐t✐♦♥s ♦♥ q✉✐❡s❝❡♥t st❛t❡s ❙✉s♣❡♥s✐♦♥ tr❛❝❡s STraces(M) Traces(∆(M)) = Traces(det(∆(M)))
s0 s1 s2 s3 s4 s5 s6 s7 ?d ?r !δ τ1 !b ?d ?r !o !a τ2 τ3 !δ τ4 !δ !δ
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❙♣❡❝✐✜❝❛t✐♦♥ ✿ ✐♦▲❚❙ S = (Q❙, A❙, − →❙, s❙
0)
■♠♣❧❡♠❡♥t❛t✐♦♥ ✿ ✐♦▲❚❙ IUT = (Q■❯❚, A■❯❚, − →■❯❚, s■❯❚
0 )
❯♥❦♥♦✇♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥✱ ❡①❝❡♣t ❢♦r ✐ts ✐♥t❡r❢❛❝❡✱ ✐❞❡♥t✐❝❛❧ t♦ S✬s ❍②♣✳✿ IUT ✐s ✐♥♣✉t✲❝♦♠♣❧❡t❡ ✿ ■♥ ❛♥② st❛t❡✱ IUT ❛❝❝❡♣ts ❛♥② ✐♥♣✉t✱ ♣♦ss✐❜❧② ❛❢t❡r ✐♥t❡r♥❛❧ ❛❝t✐♦♥s✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❤❡ ❝♦♥❢♦r♠❛♥❝❡ r❡❧❛t✐♦♥ ❞❡✜♥❡s t❤❡ s❡t ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥s IUT ❝♦♥❢♦r♠✐♥❣ t♦ S✳ ❈♦♥❢♦r♠❛♥❝❡ IUT ✐♦❝♦ S ∀σ ∈ STraces(S), Out(∆(IUT) ❛❢t❡r σ) ⊆ Out(∆(S) ❛❢t❡r σ) ✇✐t❤ Out(P) Γ(P) ∩ Aδ
O ❛✿ s❡t ♦❢ ♦✉t♣✉ts ∧ q✉✐❡s❝❡♥❝❡s ✐♥ P✳
❛ Aδ O ✐s ❡q✉✐✈❛❧❡♥t ♥♦t❛t✐♦♥ ❢♦r AO s✐♥❝❡ δ ✐s ❛♥ ♦✉t♣✉t ♦❢ ∆(S) ❛♥❞
∆(IUT)
■♥t✉✐t✐♦♥ ✿ IUT ❝♦♥❢♦r♠s t♦ S ✐✛ ❛❢t❡r ❛♥② s✉s♣❡♥s✐♦♥ tr❛❝❡ ♦❢ S ❛♥❞ IUT✱ ❛❧❧ ♦✉t♣✉ts ❛♥❞ q✉✐❡s❝❡♥❝❡s ♦❢ IUT ❛r❡ s♣❡❝✐✜❡❞ ❜② S✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
s0 s1 s2 s3 ?a !δ !x !y !z !δ
s♣❡❝✐✜❝❛t✐♦♥ ∆(S)
s0 s1 s2 ?a !δ !x !z
I1✿ ■♠♣❧❡♠✳ ❝❤♦✐❝❡
s0 s1 s2 s3 s4 ?a !δ !x !y !z !z !δ !δ
I3✿ ❯♥s♣❡❝✳ ♦✉t♣✉t
s0 s1 s2 s3 s4 s5 ?a !δ ?b !x !y !z !δ !z !δ
I2✿ ■♠♣❧❡♠✳ ♦❢ ❛ ♣❛rt✐❛❧ s♣❡❝✳
s0 s1 s2 s3 s4 ?a !δ ?a !x !y !z !δ !δ
I4 ✿ ❯♥s♣❡❝✳ q✉✐❡s❝❡♥❝❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋r♦♠ S ✭♠♦r❡ ♣r❡❝✐s❡❧② ❢r♦♠ det(∆(S)) = (Q❞, A❞, − →❞, q❞
0)✮✱
❜✉✐❧❞ ❛♥ ✐♦▲❚❙ Can(S) = (Q❝, A❝, − →❝, q❝
0)
→ t❤❡ ♠♦st ❣❡♥❡r❛❧ ✐♦▲❚❙ ❞❡t❡❝t✐♥❣ ♥♦♥✲❝♦♥❢♦r♠❛♥❝❡ ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ IUT✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋r♦♠ S ✭♠♦r❡ ♣r❡❝✐s❡❧② ❢r♦♠ det(∆(S)) = (Q❞, A❞, − →❞, q❞
0)✮✱
❜✉✐❧❞ ❛♥ ✐♦▲❚❙ Can(S) = (Q❝, A❝, − →❝, q❝
0)
→ t❤❡ ♠♦st ❣❡♥❡r❛❧ ✐♦▲❚❙ ❞❡t❡❝t✐♥❣ ♥♦♥✲❝♦♥❢♦r♠❛♥❝❡ ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ IUT✳ ❋r♦♠ det(∆(S))✮ ✿ ■♥✈❡rt ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ✭t❡st❡r ♣♦✐♥t ♦❢ ✈✐❡✇✮ ❆❧❧ ♥♦♥✲s♣❡❝✐✜❡❞ ♦✉t♣✉ts ❧❡❛❞ t♦ ❋❛✐❧✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋r♦♠ S ✭♠♦r❡ ♣r❡❝✐s❡❧② ❢r♦♠ det(∆(S)) = (Q❞, A❞, − →❞, q❞
0)✮✱
❜✉✐❧❞ ❛♥ ✐♦▲❚❙ Can(S) = (Q❝, A❝, − →❝, q❝
0)
→ t❤❡ ♠♦st ❣❡♥❡r❛❧ ✐♦▲❚❙ ❞❡t❡❝t✐♥❣ ♥♦♥✲❝♦♥❢♦r♠❛♥❝❡ ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ IUT✳
q0 q1 q2 q3 q4 ?d ?r !δ !b ?d ?r !o !a !δ !δ
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋r♦♠ S ✭♠♦r❡ ♣r❡❝✐s❡❧② ❢r♦♠ det(∆(S)) = (Q❞, A❞, − →❞, q❞
0)✮✱
❜✉✐❧❞ ❛♥ ✐♦▲❚❙ Can(S) = (Q❝, A❝, − →❝, q❝
0)
→ t❤❡ ♠♦st ❣❡♥❡r❛❧ ✐♦▲❚❙ ❞❡t❡❝t✐♥❣ ♥♦♥✲❝♦♥❢♦r♠❛♥❝❡ ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ IUT✳
q0 q1 q2 q3 q4 !d !r ?δ ?b !d !r ?o ?a ?δ ?δ
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❋r♦♠ S ✭♠♦r❡ ♣r❡❝✐s❡❧② ❢r♦♠ det(∆(S)) = (Q❞, A❞, − →❞, q❞
0)✮✱
❜✉✐❧❞ ❛♥ ✐♦▲❚❙ Can(S) = (Q❝, A❝, − →❝, q❝
0)
→ t❤❡ ♠♦st ❣❡♥❡r❛❧ ✐♦▲❚❙ ❞❡t❡❝t✐♥❣ ♥♦♥✲❝♦♥❢♦r♠❛♥❝❡ ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ IUT✳
q0 q1 q2 q3 q4 ❋❛✐❧ !d !r ?δ ?a, ?o, ?b ?b !d !r ?a, ?o, ?δ ?o ?a ?b, ?δ ?δ ?a, ?b, ?o ?δ ?a, ?b, ?o
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❆ t❡st ❝❛s❡ ✐s ❛ ❞❡t❡r♠✐♥✐st✐❝ ✐♦▲❚❙ (Q❚❈, A❚❈, − →❚❈, t❚❈
0 )✱ ❡q✉✐♣♣❡❞ ✇✐t❤
✈❡r❞✐❝t st❛t❡s✿ P❛ss✱ ❋❛✐❧ ❛♥❞ ■♥❝♦♥❝ s✳t✳ TC ❢♦❧❧♦✇s t❤❡ t❡st❡r ♣♦✐♥t ♦❢ ✈✐❡✇ ✭✐♥♣✉t ✴ ♦✉t♣✉t ✐♥✈❡rs✐♦♥✮ TC ✐s ❝♦♥tr♦❧❧❛❜❧❡✱ ✐✳❡✳ ♥❡✈❡r ❤❛✈❡ t♦ ❝❤♦♦s❡ ❜t✇✳ s❡✈❡r❛❧ ♦✉t♣✉ts ♦r ❜t✇✳ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ✿ ❆❧❧ st❛t❡s ✇✐t❤ ❛♥ ✐♥♣✉t✱ ❛r❡ ✐♥♣✉t✲❝♦♠♣❧❡t❡✱ ❡①❝❡♣t ✈❡r❞✐❝t st❛t❡s✳
t0 t1 t2 t3 t4 P❛ss ■♥❝♦♥❝ ❋❛✐❧ ?δ ❄♦t❤✇ !d !d ?o ?a ❄♦t❤✇ ?δ ❄♦t❤✇
❚❡st ❡①❡❝✉t✐♦♥ ❂ ♣❛r❛❧❧❡❧ ❝♦♠♣♦s✐t✐♦♥ TC∆(IUT) s②♥❝❤r♦♥✐③✐♥❣ ♦♥ ❝♦♠♠♦♥ ✈✐s✐❜❧❡ ❛❝t✐♦♥s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❆ t❡st ❝❛s❡ ✐s ❛ ❞❡t❡r♠✐♥✐st✐❝ ✐♦▲❚❙ (Q❚❈, A❚❈, − →❚❈, t❚❈
0 )✱ ❡q✉✐♣♣❡❞ ✇✐t❤
✈❡r❞✐❝t st❛t❡s✿ P❛ss✱ ❋❛✐❧ ❛♥❞ ■♥❝♦♥❝ s✳t✳ TC ❢♦❧❧♦✇s t❤❡ t❡st❡r ♣♦✐♥t ♦❢ ✈✐❡✇ ✭✐♥♣✉t ✴ ♦✉t♣✉t ✐♥✈❡rs✐♦♥✮ TC ✐s ❝♦♥tr♦❧❧❛❜❧❡✱ ✐✳❡✳ ♥❡✈❡r ❤❛✈❡ t♦ ❝❤♦♦s❡ ❜t✇✳ s❡✈❡r❛❧ ♦✉t♣✉ts ♦r ❜t✇✳ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ✿ ❆❧❧ st❛t❡s ✇✐t❤ ❛♥ ✐♥♣✉t✱ ❛r❡ ✐♥♣✉t✲❝♦♠♣❧❡t❡✱ ❡①❝❡♣t ✈❡r❞✐❝t st❛t❡s✳
t0 t1 t2 t3 t4 P❛ss ■♥❝♦♥❝ ❋❛✐❧ ?δ ❄♦t❤✇ !d !d ?o ?a ❄♦t❤✇ ?δ ❄♦t❤✇
❚❡st ❡①❡❝✉t✐♦♥ ❂ ♣❛r❛❧❧❡❧ ❝♦♠♣♦s✐t✐♦♥ TC∆(IUT) s②♥❝❤r♦♥✐③✐♥❣ ♦♥ ❝♦♠♠♦♥ ✈✐s✐❜❧❡ ❛❝t✐♦♥s
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
TC ❢❛✐❧s IUT ✐✛ ❛♥ ❡①❡❝✉t✐♦♥ ♦❢ TC∆(IUT) r❡❛❝❤❡s ❋❛✐❧ ❊①♣r❡ss❡s ❛ ♣♦ss✐❜✐❧✐t② ❢♦r r❡❥❡❝t✐♦♥✳ → ❛ s✐♥❣❧❡ t❡st ❝❛s❡ ♠❛② ❧❡❛❞ t♦ s❡✈❡r❛❧ ❞✐✛❡r❡♥t ✈❡r❞✐❝ts ❙♦✉♥❞♥❡ss✱ ❊①❤❛✉st✐✈❡♥❡ss✱ ❈♦♠♣❧❡t❡♥❡ss ❆ s❡t ♦❢ t❡st ❝❛s❡s TS ✐s ❙♦✉♥❞ ∀IUT : (IUT ✐♦❝♦ S = ⇒ ∀TC ∈ TS : ¬(TC fails IUT))✱ ✐✳❡✳ ♦♥❧② ♥♦♥✲❝♦♥❢♦r♠❛♥t IUT ♠❛② ❜❡ r❡❥❡❝t❡❞ ❜② ❛ TC ∈ TS✳ ❊①❤❛✉st✐✈❡ ∀IUT : (¬(IUT ✐♦❝♦ S) = ⇒ ∃TC ∈ TS : TC fails IUT)✱ ✐✳❡✳ ❛♥② ♥♦♥✲❝♦♥❢♦r♠❛♥t IUT ♠❛② ❜❡ r❡❥❡❝t❡❞ ❜② ❛ TC ∈ TS✳ ❈♦♠♣❧❡t❡ ❂ ❙♦✉♥❞ ❛♥❞ ❊①❤❛✉st✐✈❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
TC ❢❛✐❧s IUT ✐✛ ❛♥ ❡①❡❝✉t✐♦♥ ♦❢ TC∆(IUT) r❡❛❝❤❡s ❋❛✐❧ ❊①♣r❡ss❡s ❛ ♣♦ss✐❜✐❧✐t② ❢♦r r❡❥❡❝t✐♦♥✳ → ❛ s✐♥❣❧❡ t❡st ❝❛s❡ ♠❛② ❧❡❛❞ t♦ s❡✈❡r❛❧ ❞✐✛❡r❡♥t ✈❡r❞✐❝ts ❙♦✉♥❞♥❡ss✱ ❊①❤❛✉st✐✈❡♥❡ss✱ ❈♦♠♣❧❡t❡♥❡ss ❆ s❡t ♦❢ t❡st ❝❛s❡s TS ✐s ❙♦✉♥❞ ∀IUT : (IUT ✐♦❝♦ S = ⇒ ∀TC ∈ TS : ¬(TC fails IUT))✱ ✐✳❡✳ ♦♥❧② ♥♦♥✲❝♦♥❢♦r♠❛♥t IUT ♠❛② ❜❡ r❡❥❡❝t❡❞ ❜② ❛ TC ∈ TS✳ ❊①❤❛✉st✐✈❡ ∀IUT : (¬(IUT ✐♦❝♦ S) = ⇒ ∃TC ∈ TS : TC fails IUT)✱ ✐✳❡✳ ❛♥② ♥♦♥✲❝♦♥❢♦r♠❛♥t IUT ♠❛② ❜❡ r❡❥❡❝t❡❞ ❜② ❛ TC ∈ TS✳ ❈♦♠♣❧❡t❡ ❂ ❙♦✉♥❞ ❛♥❞ ❊①❤❛✉st✐✈❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❖❜❥❡❝t✐✈❡ ✿ ❋✐♥❞ ❛♥ ❛❧❣♦r✐t❤♠ t❛❦✐♥❣ ❛s ✐♥♣✉t ❛ ✜♥✐t❡ st❛t❡ ✐♦▲❚❙ S✱ ❛♥❞ s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ Pr♦❞✉❝❡s ♦♥❧② s♦✉♥❞ t❡st s✉✐t❡s ■s ❧✐♠✐t✲❡①❤❛✉st✐✈❡ ✐✳❡✳ t❤❡ ✐♥✜♥✐t❡ s✉✐t❡ ♦❢ t❡st ❝❛s❡s t❤❛t ❝❛♥ ❜❡ ♣r♦❞✉❝❡❞ ✐s ❡①❤❛✉st✐✈❡ ❚✇♦ t❡❝❤♥✐q✉❡s ✿
✶ ◆♦♥✲❞❡t❡r♠✐♥✐st✐❝ s❡❧❡❝t✐♦♥ ✭❚♦r❳✮ ✷ ❙❡❧❡❝t✐♦♥ ❣✉✐❞❡❞ ❜② ❛ t❡st ♣✉r♣♦s❡ ✭❚●❱✮ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✽✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❆❧❣♦r✐t❤♠✿ ♣❛rt✐❛❧ ✉♥❢♦❧❞✐♥❣ ♦❢ Can(S) ❙t❛rt ✐♥ qc
0✳ ❆❢t❡r ❛♥② tr❛❝❡ σ ✐♥ Can(S)
■❢ Can(S) ❛❢t❡r σ ⊆ ❋❛✐❧✱ ❡♠✐t ❛ ❋❛✐❧ ✈❡r❞✐❝t ❖t❤❡r✇✐s❡ ♠❛❦❡ ❛ ❝❤♦✐❝❡ ❜❡t✇❡❡♥
Pr♦❞✉❝❡ ❛ P❛ss ✈❡r❞✐❝t ❛♥❞ st♦♣✱ ❈♦♥s✐❞❡r ❛❧❧ ✐♥♣✉ts ♦❢ Can(S) ❛❢t❡r σ ❛♥❞ ❝♦♥t✐♥✉❡✳ ❈❤♦♦s❡ ♦♥❡ ♦✉t♣✉t ✐♥ t❤♦s❡ ♦❢ Can(S) ❛❢t❡r σ ❛♥❞ ❝♦♥t✐♥✉❡✳
Pr♦♣❡rt✐❡s TS ❂ ❛❧❧ ♣♦ss✐❜❧❡ ❚❡st ❝❛s❡s ❣❡♥❡r❛t❡❞ ✇✐t❤ t❤✐s ❛❧❣♦r✐t❤♠ ✿ TS ✐s s♦✉♥❞ ❛♥❞ ❧✐♠✐t✲❡①❤❛✉st✐✈❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❆❧❣♦r✐t❤♠✿ ♣❛rt✐❛❧ ✉♥❢♦❧❞✐♥❣ ♦❢ Can(S) ❙t❛rt ✐♥ qc
0✳ ❆❢t❡r ❛♥② tr❛❝❡ σ ✐♥ Can(S)
■❢ Can(S) ❛❢t❡r σ ⊆ ❋❛✐❧✱ ❡♠✐t ❛ ❋❛✐❧ ✈❡r❞✐❝t ❖t❤❡r✇✐s❡ ♠❛❦❡ ❛ ❝❤♦✐❝❡ ❜❡t✇❡❡♥
Pr♦❞✉❝❡ ❛ P❛ss ✈❡r❞✐❝t ❛♥❞ st♦♣✱ ❈♦♥s✐❞❡r ❛❧❧ ✐♥♣✉ts ♦❢ Can(S) ❛❢t❡r σ ❛♥❞ ❝♦♥t✐♥✉❡✳ ❈❤♦♦s❡ ♦♥❡ ♦✉t♣✉t ✐♥ t❤♦s❡ ♦❢ Can(S) ❛❢t❡r σ ❛♥❞ ❝♦♥t✐♥✉❡✳
Pr♦♣❡rt✐❡s TS ❂ ❛❧❧ ♣♦ss✐❜❧❡ ❚❡st ❝❛s❡s ❣❡♥❡r❛t❡❞ ✇✐t❤ t❤✐s ❛❧❣♦r✐t❤♠ ✿ TS ✐s s♦✉♥❞ ❛♥❞ ❧✐♠✐t✲❡①❤❛✉st✐✈❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
q0 q1 q2 q3 q4 ❋❛✐❧ !d !r ?δ ?a, ?o, ?b ?b !d !r ?a, ?o, ?δ ?o ?a ?b, ?δ ?δ ?a, ?b, ?o ?δ ?a, ?b, ?o
q0 q0 q0 q1 q4 ❋❛✐❧ ❋❛✐❧ ❋❛✐❧ P❛ss !r ?δ ?a, ?b, ?o !d ?b ?a, ?o, ?δ ?δ ?a, ?o, ?b
Can(S) TC1
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
Pr❡✈✐♦✉s ❛❧❣♦r✐t❤♠ ✿ ♠❛②❜❡ q✉✐t❡ ❧♦♥❣ ✐❢ ✇❡ ✐♥t❡♥❞ t♦ ❢♦❝✉s ♦♥ ❛ s♣❡❝✐✜❝ ❜❡❤❛✈✐♦r✳✳✳ ▼❛✐♥ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❚❡st P✉r♣♦s❡ ●❡♥❡r❛t✐♦♥✿ ❚❡st s❡❧❡❝t✐♦♥ ❜② t❡st ♣✉r♣♦s❡s ❞❡s❝r✐❜✐♥❣ ❛ s❡t ♦❢ ❜❡❤❛✈✐♦rs t♦ ❜❡ t❡st❡❞✱ t❛r❣❡t❡❞ ❜② ❛ t❡st ❝❛s❡
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚❡st P✉r♣♦s❡ ❉❡t❡r♠✐♥✐st✐❝ ❛♥❞ ❝♦♠♣❧❡t❡ ✐♦▲❚❙ TP = (Q❚P, A❚P, − →❚P, q❚P
0 )
❡q✉✐♣♣❡❞ ✇✐t❤ t✇♦ s❡ts Accept❚P ❛♥❞ Refuse❚P ♦❢ tr❛♣ st❛t❡s
q0 q1 q2 q3 q4 ❋❛✐❧ !d !r ?δ ?a, ?o, ?b ?b !d !r ?a, ?o, ?δ ?o ?a ?b, ?δ ?δ ?a, ?b, ?o ?δ ?a, ?b, ?o p0 p1 p2 Accept p3 Refuse ?r ∗ !o ?r ∗ ∗ ∗
Can(S) T P ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✸✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❉❡✜♥✐t✐♦♥ ♦❢ ❙②♥❝❤r♦♥♦✉s Pr♦❞✉❝t ❚❤❡ ❙②♥❝❤r♦♥♦✉s Pr♦❞✉❝t ♦❢ t✇♦ ✐♦▲❚❙ M1 = (Q▼✶, A, − →▼✶, q▼✶
0 )✱ ❛♥❞ M2 = (Q▼✷, A, −
→▼✷, q▼✷
0 ) ✐s t❤❡
✐♦▲❚❙ M1 × M2 = (Q▼✶ × Q▼✷, A, − →, q▼✶
0 × q▼✷ 0 ) ✇❤❡r❡ −
→ ✐s ❞❡✜♥❡❞ ❜② ✿ (q▼✶, q▼✷)
a
− → (q′
▼✶, q′ ▼✷) ⇔ (q▼✶
a
− →▼✶ q′
▼✶) ∧ (q▼✷
a
− →▼✷ q′
▼✷) ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✹✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
q0 q1 q2 q3 q4 ❋❛✐❧ !d !r ?δ ?a, ?o, ?b ?b !d !r ?a, ?o, ?δ ?o ?a ?b, ?δ ?δ ?a, ?b, ?o ?δ ?a, ?b, ?o p0 p1 p2 Accept p3 Refuse ?r ∗ !o ?r ∗ ∗ ∗
q0✱p0 q0✱p1 q1✱p0 q0✱p3 q1✱p1 q4✱p0 q2✱p0 q1✱p3 q2✱p1 q4✱p1 q3✱p0 q4✱p3 q2✱p3 q0✱p2 q3✱p1 q3✱p3 q1✱p2 Fail !r ?δ !d !r ?δ !d ?b !d !r !d ?δ✱ !r !r !d ?b ?δ ?a ?o ?b !d !r ?o ?a ?δ ?δ ?δ ?a ?o !d ?δ !r ?δ ?δ ?othw
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✺✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❈♦✲r❡❛❝❤❛❜✐❧✐t② ❛♥❛❧②s✐s ✿ ❑❡❡♣ t❤❡ ✜rst Accept st❛t❡ ✐♥ ❛ ♣❛t❤ → P❛ss ■❢ q ∈ coreach(Pass) ❦❡❡♣ q ■❢ q ∈ {❋❛✐❧} ❦❡❡♣ q ■❢ q ∈ coreach(Pass) ✐♥♣✉t ✭t❡st❡r ♣♦✐♥t ♦❢ ✈✐❡✇✮ ✐s s✉❝❝❡ss♦r ♦❢ ❛ st❛t❡ q′ ∈ coreach(Pass) t❤❡♥ Inconc
q0, p0 q0, p1 q1, p0 q1, p1 ■♥❝♦♥❝ q2, p0 q2, p1 ■♥❝♦♥❝ ■♥❝♦♥❝ P❛ss ■♥❝♦♥❝ Fail !r ?δ !d ?δ !d ?b !d !r !d ?b ?a ?o ?o ?a ?othw
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✻✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
q0, p0 q0, p1 q0, p1 q1, p1 q2, p1 P❛ss ■♥❝♦♥❝ ❋❛✐❧ !r ?δ !d !d ?o ?a ?othw ?othw
❊①❛♠♣❧❡ ♦❢ ❚❡st ❈❛s❡
q0, p0 q0, p1 q1, p0 q1, p1 ■♥❝♦♥❝ q2, p0 q2, p1 ■♥❝♦♥❝ ■♥❝♦♥❝ P❛ss ■♥❝♦♥❝ Fail !r ?δ !d ?δ !d ?b !d !r !d ?b ?a ?o ?o ?a ?othw
❚❤❡ t❡st s✉✐t❡ ❝♦♠♣♦s❡❞ ♦❢ t❤❡ s❡t ♦❢ t❡st ❝❛s❡s t❤❛t t❤❡ ❛❧❣♦r✐t❤♠ ❝❛♥ ♣r♦❞✉❝❡ ✐s s♦✉♥❞ ❛♥❞ ❧✐♠✐t✲❡①❤❛✉st✐✈❡✳
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✼✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❚✇♦ ✏❤✐st♦r✐❝❛❧✑ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ✐♦▲❚❙ ❛♥❞ ▼❡❛❧② ▼❛❝❤✐♥❡s t❤❡♦r② ❚♦❞❛②✱ ♠❛♥② ♦t❤❡r ❛♣♣r♦❛❝❤❡s ❡①✐st✱ ✇✐t❤ ✈❛r✐♦✉s ❞❡s❝r✐❜✐♥❣ ❢♦r♠❛ts ✭❡✳❣✳ ❡①t❡♥s✐♦♥s ♦❢ ❋❙▼ ❛♥❞ ▲❚❙✱ ❯▼▲✱ ❙②s▼▲✱ ▼❛r❦♦✈ ❝❤❛✐♥s✱ ❙✐♠✉❧✐♥❦✱ ▲✉str❡✱ ✳✳✳✮ ▼❛♥② t♦♦❧s ❛r❡ ❛✈❛✐❧❛❜❧❡✳ → ❆ ✭♥♦♥✲❡①❤❛✉st✐✈❡✱ ❜✉t ②❡t ✐♥t❡r❡st✐♥❣✮ ❧✐st ♠❛② ❜❡ ❢♦✉♥❞ ❤❡r❡ ✿
❤tt♣✿✴✴♠✐t✳❜♠❡✳❤✉✴⑦♠✐❝s❦❡✐③✴♣❛❣❡s✴♠♦❞❡❧❜❛s❡❞❴t❡st✐♥❣✳❤t♠❧
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✽✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
✶ ●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ✷ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ✸ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣
✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡s ✧❍✐st♦r✐❝❛❧✧ ❛♣♣r♦❛❝❤❡s ♦❢ ▼❇❚ ✿ ❜❛s❡❞ ♦♥ ▲❛❜❡❧❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s
✹ ❈♦♥❝❧✉s✐♦♥
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✻✾✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❱❡r② ❛❝t✐✈❡ ❞♦♠❛✐♥ ▼❛♥② ✐ss✉❡s✱ ❜♦t❤ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♣r❛❝t✐❝❛❧ ❙t✐❧❧ ❛ ❧♦t t♦ ❞♦ ❍✉❣❡ ✐♥❞✉str✐❛❧ ♥❡❡❞s P❡rs♣❡❝t✐✈❡s ▼❛♥② ✦ ❙❡❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤✐s ❙✉♠♠❡r ❙❝❤♦♦❧ ✦
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✼✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❱❡r② ❛❝t✐✈❡ ❞♦♠❛✐♥ ▼❛♥② ✐ss✉❡s✱ ❜♦t❤ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♣r❛❝t✐❝❛❧ ❙t✐❧❧ ❛ ❧♦t t♦ ❞♦ ❍✉❣❡ ✐♥❞✉str✐❛❧ ♥❡❡❞s P❡rs♣❡❝t✐✈❡s ▼❛♥② ✦ ❙❡❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤✐s ❙✉♠♠❡r ❙❝❤♦♦❧ ✦
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✼✵✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
r♦❧❧❡t❅❧❛❜r✐✳❢r
❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✼✶✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❬●❑❙✵✺❪ P✳ ●♦❞❡❢r♦✐❞✱ ◆✳ ❑❧❛r❧✉♥❞✱ ❛♥❞ ❑✳ ❙❡♥✳ ✏❉❛rt✿ ❞✐r❡❝t❡❞ ❛✉t♦♠❛t❡❞ r❛♥❞♦♠ t❡st✐♥❣✑✱ ❙■●P▲❆◆ ◆♦t✳✱ ✹✵✭✻✮✿✷✶✸✕✷✷✸✱ ✷✵✵✺✳ ❬❙▼❆✵✺❪ ❑✳ ❙❡♥✱ ❉✳ ▼❛r✐♥♦✈✱ ❛♥❞ ●✳ ❆❣❤❛✳ ✏❈✉t❡✿ ❛ ❝♦♥❝♦❧✐❝ ✉♥✐t t❡st✐♥❣ ❡♥❣✐♥❡ ❢♦r ❝✑✱ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✶✵t❤ ❊✉r♦♣❡❛♥ s♦❢t✇❛r❡ ❡♥❣✐♥❡❡r✐♥❣ ❝♦♥❢❡r❡♥❝❡ ✱ ❊❙❊❈✴❋❙❊✲✶✸✱ ♣❛❣❡s ✷✻✸✕✷✼✷✳ ❆❈▼✱ ✷✵✵✺✳ ❬●♦t✵✾❪ ❆✳ ●♦t❧✐❡❜✳ ✏❊✉❝❧✐❞❡✿ ❆ ❝♦♥str❛✐♥t✲❜❛s❡❞ t❡st✐♥❣ ♣❧❛t❢♦r♠ ❢♦r ❝r✐t✐❝❛❧ ❝ ♣r♦❣r❛♠s✑✱ ■♥ ✷t❤ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙♦❢t✇❛r❡ ❚❡st✐♥❣✱ ❱❛❧✐❞❛t✐♦♥ ❛♥❞ ❱❡r✐✜❝❛t✐♦♥ ✭■❈❙❚✬✵✾✮✱ ❉❡♥✈❡r✱ ❈❖✱ ♣❛❣❡ ✶✵♣✱ ✵✹ ✷✵✵✾✳ ❬❈●❏✰✵✵❪ ❊✳ ❈❧❛r❦❡✱ ❖✳ ●r✉♠❜❡r❣✱ ❙✳ ❏❤❛✱ ❨✳ ▲✉✱ ❛♥❞ ❍✳ ❱❡✐t❤✳ ✏❈♦✉♥t❡r❡①❛♠♣❧❡✲❣✉✐❞❡❞ ❛❜str❛❝t✐♦♥ r❡✜♥❡♠❡♥t✑✱ ■♥ ❊✳ ❊♠❡rs♦♥ ❛♥❞ ❆✳ ❙✐st❧❛✱ ❡❞✐t♦rs✱ ❈♦♠♣✉t❡r ❆✐❞❡❞ ❱❡r✐✜❝❛t✐♦♥✱ ✈♦❧✉♠❡ ✶✽✺✺ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✶✺✹✕✶✻✾✳ ❙♣r✐♥❣❡r✱ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✵✵✳ ❬❲▼▼❘✵✺❪ ◆ ✳❲✐❧❧✐❛♠s✱ ❇✳ ▼❛rr❡✱ P✳ ▼♦✉②✱ ❛♥❞ ▼✳ ❘♦❣❡r✳ ✏P❛t❤❝r❛✇❧❡r✿ ❆✉t♦♠❛t✐❝ ❣❡♥❡r❛t✐♦♥ ♦❢ ♣❛t❤ t❡sts ❜② ❝♦♠❜✐♥✐♥❣ st❛t✐❝ ❛♥❞ ❞②♥❛♠✐❝ ❛♥❛❧②s✐s✑✱ ■♥ ❊❉❈❈✱ ✈♦❧✉♠❡ ✸✹✻✸ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✷✽✶✕✷✾✷✳ ❙♣r✐♥❣❡r✱ ✷✵✵✺✳ ❬❇❍✵✽❪ ❙✳ ❇❛r❞✐♥ ❛♥❞ P✳ ❍❡rr♠❛♥♥✳ ✏❙tr✉❝t✉r❛❧ t❡st✐♥❣ ♦❢ ❡①❡❝✉t❛❜❧❡s✳✑ ■♥ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙♦❢t✇❛r❡ ❚❡st✐♥❣✱ ❱❡r✐✜❝❛t✐♦♥✱ ❛♥❞ ❱❛❧✐❞❛t✐♦♥✱ ▲♦s ❆❧❛♠✐t♦s✱ ❈❆✱ ❯❙❆✱ ♣❛❣❡s ✷✷✕✸✶✳ ■❊❊❊ ❈♦♠♣✉t❡r ❙♦❝✐❡t②✱ ✷✵✵✽✳ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✼✷✴✼✸
❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥
❬▲❨✾✻❪ ❉✳ ▲❡❡ ❛♥❞ ▼✳ ❨❛♥♥❛❦❛❦✐s✳✑Pr✐♥❝✐♣❧❡s ❛♥❞ ♠❡t❤♦❞s ♦❢ t❡st✐♥❣ ✜♥✐t❡ st❛t❡ ♠❛❝❤✐♥❡s ✲ ❛ s✉r✈❡②✑✱ Pr♦❝✳ ♦❢ t❤❡ ■❊❊❊✱ ✽✹✿✶✵✾✵✕✶✶✷✸✱ ✽ ✶✾✾✻✳ ❬P❡t✵✵❜❪ ❆✳ P❡tr❡♥❦♦✳✑❋❛✉❧t ♠♦❞❡❧✲❞r✐✈❡♥ t❡st ❞❡r✐✈❛t✐♦♥ ❢r♦♠ ✜♥✐t❡ st❛t❡ ♠♦❞❡❧s✿ ❆♥♥♦t❛t❡❞ ❜✐❜❧✐♦❣r❛♣❤②✑✱ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✹t❤ ❙✉♠♠❡r ❙❝❤♦♦❧ ♦♥ ▼♦❞❡❧✐♥❣ ❛♥❞ ❱❡r✐✜❝❛t✐♦♥ ♦❢ P❛r❛❧❧❡❧ Pr♦❝❡ss❡s✱ ▼❖❱❊P ✬✵✵✱ ♣❛❣❡s ✶✾✻✕✷✵✺✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✷✵✵✵✳ ❬◆❚✽✶❪ ❙✳ ◆❛✐t♦ ❛♥❞ ▼✳ ❚s✉♥♦②❛♠❛✳ ✏❋❛✉❧t ❉❡t❡❝t✐♦♥ ❢♦r ❙❡q✉❡♥t✐❛❧ ▼❛❝❤✐♥❡s ❜② ❚r❛♥s✐t✐♦♥✲❚♦✉rs✑✱ Pr♦❝❡❡❞✐♥❣s ♦❢ ❋❛✉❧t ❚♦❧❡r❛♥t ❈♦♠♣✉t❡r ❙②st❡♠s✱ ♣❛❣❡s ✷✸✽✕✷✹✸✱ ✶✾✽✶✳ ❬●♦♥✼✵❪
❈♦♠♣✉t❡rs✱ ❈✲✶✾✿✺✺✶✕✺✺✽✱ ✶✾✼✵✳ ❬❙❉✽✽❪ ❑✳ ❙❛❜♥❛♥✐ ❛♥❞ ❆✳ ❉❛❤❜✉r❛✳ ✏❆ ♣r♦t♦❝♦❧ t❡st ❣❡♥❡r❛t✐♦♥ ♣r♦❝❡❞✉r❡✑✱ ❈♦♠♣✉t❡r ◆❡t✇♦r❦s ❛♥❞ ■❙❉◆ ❙②st❡♠s✱ ✶✺✿✷✽✺✕✷✾✼✱ ✶✾✽✽✳ ❬❈❤♦✼✽❪ ❚✳❙✳ ❈❤♦✇✳ ✏❚❡st✐♥❣ s♦❢t✇❛r❡ ❞❡s✐❣♥ ♠♦❞❡❧❡❞ ❜② ✜♥✐t❡✲st❛t❡ ♠❛❝❤✐♥❡s✑✱ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❙♦❢t✇❛r❡ ❊♥❣✐♥❡❡r✐♥❣✱ ❙❊✲✹✭✸✮✿✶✼✽✕✶✽✼✱ ✶✾✼✽✳ ❬❱❈■✽✾❪ ❙✳ ❱✉♦♥❣✱ ❲✳ ❈❤❛♥✱ ❛♥❞ ▼✳ ■t♦✳ ✏❚❤❡ ❯■❖✈✲▼❡t❤♦❞ ❢♦r Pr♦t♦❝♦❧ ❚❡st ❙❡q✉❡♥❝❡ ●❡♥❡r❛t✐♦♥✑✱ ■♥ ✷♥❞ ■❲P❚❙ ■♥t❡r♥❛t✐♦♥❛❧ ❲♦r❦s❤♦♣ ♦♥ Pr♦t♦❝♦❧ ❚❡st ❙②st❡♠s✱ ❇❡r❧✐♥✱ ✶✾✽✾✳ ❬❋❇❑✰✾✶❪ ❙✳ ❋✉❥✐✇❛r❛✱ ●✳ ❇♦❝❤♠❛♥♥✱ ❋✳ ❑❤❡♥❞❡❦✱ ▼✳ ❆♠❛❧♦✉✱ ❛♥❞ ❆✳ ●❤❡❞❛♠s✐✳ ✏❚❡st s❡❧❡❝t✐♦♥ ❜❛s❡❞ ♦♥ ✜♥✐t❡✲st❛t❡ ♠♦❞❡❧s✑✱ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❙♦❢t✇❛r❡ ❊♥❣✐♥❡❡r✐♥❣✱ ✶✼✭✻✮✿✺✾✶✕✻✵✸✱ ❏✉♥❡ ✶✾✾✶✳ ❬❈❆✾✷❪ ❲✳ ❈❤✉♥❣ ❛♥❞ P✳ ❆♠❡r✳ ✏■♠♣r♦✈❡❞ ♦♥ ❯■❖ ❙❡q✉❡♥❝❡ ●❡♥❡r❛t✐♦♥ ❛♥❞ P❛rt✐❛❧ ❯■❖ ❙❡q✉❡♥❝❡s✑✱ ■♥ Pr♦t♦❝♦❧ ❙♣❡❝✐✜❝❛t✐♦♥✱ ❚❡st✐♥❣✱ ❛♥❞ ❱❡r✐✜❝❛t✐♦♥✱ ❳■■✱ ▲❛❦❡ ❇✉❡♥❛ ❱✐st❛✱ ❋❧♦r✐❞❛✱ ❯❙❆✳ ◆♦rt❤✲❍♦❧❧❛♥❞✱ ❏✉♥❡ ✶✾✾✷✳ ❬❇❚✵✵❪ ❊✳ ❇r✐♥❦s♠❛ ❛♥❞ ❏✳ ❚r❡t♠❛♥s✳ ✏❚❡st✐♥❣ ❚r❛♥s✐t✐♦♥ ❙②st❡♠s✿ ❆♥ ❆♥♥♦t❛t❡❞ ❇✐❜❧✐♦❣r❛♣❤②✑✳ ❙✉♠♠❡r ❙❝❤♦♦❧ ▼❖❱❊P✬✷❦ ✕ ▼♦❞❡❧❧✐♥❣ ❛♥❞ ❱❡r✐✜❝❛t✐♦♥ ♦❢ P❛r❛❧❧❡❧ Pr♦❝❡ss❡s✱ ♣✳ ✹✹✕✺✵✱ ◆❛♥t❡s✱ ✷✵✵✵✳ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✼✸✴✼✸