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●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❉✐✣❝✉❧t②✭✐❡s✮ ♦❢ t❡st✐♥❣ ❚❡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 ❥✮❀ 2 64 ♣♦ss✐❜✐❧✐t✐❡s✳ ❈♦♥s✐❞❡r✐♥❣ ♦♥❡ t❡st ♣❡r ♠✐❝r♦✲s❡❝♦♥❞ → ✺✽✸✵✵✵ ②❡❛rs✳✳✳ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✾✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❉✐✣❝✉❧t②✭✐❡s✮ ♦❢ t❡st✐♥❣ ❚❡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 ❥✮❀ 2 64 ♣♦ss✐❜✐❧✐t✐❡s✳ ❈♦♥s✐❞❡r✐♥❣ ♦♥❡ t❡st ♣❡r ♠✐❝r♦✲s❡❝♦♥❞ → ✺✽✸✵✵✵ ②❡❛rs✳✳✳ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✾✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❉✐✣❝✉❧t②✭✐❡s✮ ♦❢ t❡st✐♥❣ ✭✷✮ ❉✐❥❦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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❉✐✣❝✉❧t②✭✐❡s✮ ♦❢ t❡st✐♥❣ ✭✷✮ ❉✐❥❦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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st✐♥❣ ♣♦✐♥t ♦❢ ✈✐❡✇ ❚❤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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st✐♥❣ ♣♦✐♥t ♦❢ ✈✐❡✇ ❚❤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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st✐♥❣ ♣♦✐♥t ♦❢ ✈✐❡✇ ❚❤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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❙♦♠❡ ✇♦r❞s ❛❜♦✉t ✐♥t❡❣r❛t✐♦♥ t❡st✐♥❣ ■♥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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❆♥♦t❤❡r ✇❛② t♦ ❝❧❛ss✐❢②✴♣♦✐♥t ♦❢ ✈✐❡✇ ❋✉♥❝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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❆♥♦t❤❡r ✇❛② t♦ ❝❧❛ss✐❢②✴♣♦✐♥t ♦❢ ✈✐❡✇ ❋✉♥❝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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❆♥❞ ❜❡②♦♥❞ t❤✐s ❝❧❛ss✐✜❝❛t✐♦♥✱ ✐♥ ❛ ❧♦♦s❡ ✇❛② ▼✉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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❆♥❞ ❜❡②♦♥❞ t❤✐s ❝❧❛ss✐✜❝❛t✐♦♥✱ ✐♥ ❛ ❧♦♦s❡ ✇❛② ▼✉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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❖✉t❧✐♥❡ ✶ ●❡♥❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❘❡❢❡r❡♥❝❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧ Pr✐♥❝✐♣❧❡ ✲ ❈♦❞❡ ❝♦✈❡r❛❣❡ ❙♦✉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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥tr♦❧ ❋❧♦✇ ●r❛♣❤ ❈❋● ❉✐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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥tr♦❧ ❋❧♦✇ ●r❛♣❤ ❈❋● ❉✐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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥tr♦❧ ❋❧♦✇ ●r❛♣❤ ✿ ❡①❛♠♣❧❡ ❦❂✵ ✴✴ ♣r♦❞✉❝t ♦❢ t✇♦ ✐♥t ✐♥t ♣r♦❞✭✐♥t ✐✱ ✐♥t ❥✮ ④ ✐♥t ❦❂✵❀ ✐❢ ✭✐ ❂❂ ✷✮ ❦ ❂ ✐❁❁✶❀ ✴✴ ❡rr♦r ❤❡r❡❀ s❤♦✉❧❞ ❜❡ ❥❁❁✶ ✐❂❂✷ ✐✦❂✷ ❡❧s❡ ④ ✇❤✐❧❡ ✭✐❃✵✮ ④ ❦❂✐❁❁✶ ❦ ❂ ❦✰❥❀ ✐✲✲❀ ⑥ ✐❁❂✵ ⑥ ✐❃✵ r❡t✉r♥ ❦❀ r❡t✉r♥ ❦ ⑥ ❦❂❦✰❥ ✐❂✐✲✶ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✶✾✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈❧❛ss✐❝❛❧ ❝♦✈❡r❛❣❡ ❝r✐t❡r✐❛ ❈♦✈❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈❧❛ss✐❝❛❧ ❝♦✈❡r❛❣❡ ❝r✐t❡r✐❛ ❈♦✈❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥str❛✐♥t ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥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✐♦♥ ●✐✈❡♥ ❛ ❧♦❝❛t✐♦♥ ✐♥ t❤❡ ♣r♦❣r❛♠ ✉♥❞❡r t❡st✱ ❛✉t♦♠❛t✐❝❛❧❧② ❣❡♥❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥str❛✐♥t ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥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✐♦♥ ●✐✈❡♥ ❛ ❧♦❝❛t✐♦♥ ✐♥ t❤❡ ♣r♦❣r❛♠ ✉♥❞❡r t❡st✱ ❛✉t♦♠❛t✐❝❛❧❧② ❣❡♥❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ P❛t❤ Pr❡❞✐❝❛t❡ P❛t❤ ♣r❡❞✐❝❛t❡ ●✐✈❡♥ ❛ ♣❛t❤ Π ♦❢ ❛ ♣r♦❣r❛♠✱ ❛ ❢♦r♠✉❧❛ ϕ Π ✐s ❛ ♣❛t❤ ♣r❡❞✐❝❛t❡ ♦❢ Π ✐❢ ❢♦r ❛ ❣✐✈❡♥ s❡t ♦❢ ✈❛❧✉❡s V ✱ V | ⇒ t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ t❤❡ = ϕ Π = ♣r♦❣r❛♠ ♦♥ V ❢♦❧❧♦✇s Π → ❋✐♥❞ ❛ s♦❧✉t✐♦♥ ✭✐❢ ❛♥②✮ t♦ ϕ Π ✐♥ ♦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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ P❛t❤ ♣r❡❞✐❝❛t❡ ❡①❛♠♣❧❡ ✶ 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 ✿ y 1 = y 0 + 2 ∧ x 1 = y 1 + 4 ∧ x 1 > 2 ∗ z 0 ∧ x 2 = y 1 + 2 ✳ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✐♥♣✉ts✱ ✇❡ ❤❛✈❡ y 0 + 6 > 2 ∗ z 0 ❊①❛♠♣❧❡ ♦❢ ❚❉ ✿ y = 0 , z = 0 ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ P❛t❤ ♣r❡❞✐❝❛t❡ ❡①❛♠♣❧❡ ✶ 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 ✿ y 1 = y 0 + 2 ∧ x 1 = y 1 + 4 ∧ x 1 > 2 ∗ z 0 ∧ x 2 = y 1 + 2 ✳ ❈♦♥s✐❞❡r✐♥❣ t❤❡ ✐♥♣✉ts✱ ✇❡ ❤❛✈❡ y 0 + 6 > 2 ∗ z 0 ❊①❛♠♣❧❡ ♦❢ ❚❉ ✿ y = 0 , z = 0 ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ✐❢✴t❤❡♥ ❀ Φ ✲❡①♣r❡ss✐♦♥s✱ ❙❙❆ ❢♦r♠ ✐❢ ✭❛✵ ❃ ✵✮ ✐❢ ✭❛❃✵✮ ✲❃ ✐❢ ✭❛✵ ❃ ✵✮ ✐❂✹❀ ✲❃ ✐✶❂✹❀ ❡❧s❡ ✲❃ ❡❧s❡ ✐❂✺❀ ✲❃ ✐✷❂✺❀ ✐✶❂✹ ✐✷❂✺ ✲❃ ✐✸❂♣❤✐✭✐✶ ✱✐✷✮❀ ✐✸❂♣❤✐✭✐✶✱✐✷✮❀ Φ ✲❡①♣r → ❞❡❝✐❞❡ t❤❡ ✈❛❧✉❡ ♦❢ i 3 ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣❛t❤ ✉s❡❞ t♦ r❡❛❝❤ ✐t a 0 > 0 = ⇒ i 1 = 4 ∧ i 3 = i 1 ¬ ( a 0 > 0) = ⇒ i 2 = 5 ∧ i 3 = i 2 ❈❤♦✐❝❡ ♦❢ ♣❛t❤ ✭ ❥♦✐♥ ♦♣❡r❛t♦r✮ ❥♦✐♥ ( a 0 > 0 ∧ i 1 = 4 ∧ i 3 = i 1 , ¬ ( a 0 > 0) ∧ i 2 = 5 ∧ i 3 = i 2 ) ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✹✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❲❤✐❧❡❀ Φ ✲❡①♣r❡ss✐♦♥s✱ ❙❙❆ ❢♦r♠ ❲❤✐❧❡ ✿ Φ ✲❡①♣r ❛❞❞❡❞ ❥✉st ❜❡❢♦r❡ t❤❡ ❞❡❝✐s✐♦♥ ■♥✐t✐❛❧ ❝♦❞❡ ❙❙❆ ❝♦❞❡ ① ❂ ✶❀ ①✶ ❂ ✶❀ ①✷ ❂ ♣❤✐✭①✶ ✱ ①✸✮❀ ✇❤✐❧❡ ✭ ① ✦❂ ✶✵ ✮ ④ ✇❤✐❧❡ ✭①✷ ✦❂ ✶✵✮ ④ ❝ ❂ ①❀ ❝ ❂ ①✷❀ ① ❂ ① ✰ ✶❀ ①✸ ❂ ①✷ ✰ ✶❀ ⑥ ⑥ ♣r✐♥t✭①✮❀ ♣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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❙②♠❜♦❧✐❝ ❉❡♣t❤ ❋✐rst ❙❡❛r❝❤ ✭❉❋❙✮ P❛t❤ ❇❛s❡❞ ❚❉ ❣❡♥❡r❛t✐♦♥ ✶ ❙❡❧❡❝t ✭❛♥♦t❤❡r✮ ♣❛t❤ Π ♦❢ t❤❡ ❈❋● ✷ ❇✉✐❧❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r❡❞✐❝❛t❡ ϕ Π ✸ ❙♦❧✈❡ ϕ Π ✭✐❢ ♣♦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♦❛❝❤ ❈♦♥str❛✐♥t s♦❧✈✐♥❣✱ ❡✈❡♥ ♦♥ ❛ s✐♥❣❧❡ ♣❛t❤✱ ♠❛② ❜❡ ❝♦st❧② � ✭✉♥✇✐♥❞✐♥❣✱ ✉♥❢❡❛s✐❜❧❡ ♣❛t❤s✱ ✳✳✳✮✳ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✻✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❙②♠❜♦❧✐❝ ❉❡♣t❤ ❋✐rst ❙❡❛r❝❤ ✭❉❋❙✮ P❛t❤ ❇❛s❡❞ ❚❉ ❣❡♥❡r❛t✐♦♥ ✶ ❙❡❧❡❝t ✭❛♥♦t❤❡r✮ ♣❛t❤ Π ♦❢ t❤❡ ❈❋● ✷ ❇✉✐❧❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r❡❞✐❝❛t❡ ϕ Π ✸ ❙♦❧✈❡ ϕ Π ✭✐❢ ♣♦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♦❛❝❤ ❈♦♥str❛✐♥t s♦❧✈✐♥❣✱ ❡✈❡♥ ♦♥ ❛ s✐♥❣❧❡ ♣❛t❤✱ ♠❛② ❜❡ ❝♦st❧② � ✭✉♥✇✐♥❞✐♥❣✱ ✉♥❢❡❛s✐❜❧❡ ♣❛t❤s✱ ✳✳✳✮✳ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✻✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❙②♠❜♦❧✐❝ ❉❡♣t❤ ❋✐rst ❙❡❛r❝❤ ✭❉❋❙✮ P❛t❤ ❇❛s❡❞ ❚❉ ❣❡♥❡r❛t✐♦♥ ✶ ❙❡❧❡❝t ✭❛♥♦t❤❡r✮ ♣❛t❤ Π ♦❢ t❤❡ ❈❋● ✷ ❇✉✐❧❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r❡❞✐❝❛t❡ ϕ Π ✸ ❙♦❧✈❡ ϕ Π ✭✐❢ ♣♦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♦❛❝❤ ❈♦♥str❛✐♥t s♦❧✈✐♥❣✱ ❡✈❡♥ ♦♥ ❛ s✐♥❣❧❡ ♣❛t❤✱ ♠❛② ❜❡ ❝♦st❧② � ✭✉♥✇✐♥❞✐♥❣✱ ✉♥❢❡❛s✐❜❧❡ ♣❛t❤s✱ ✳✳✳✮✳ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✷✻✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥❝♦❧✐❝ ❛♣♣r♦❛❝❤ ■❞❡❛ ✿ ❛❝❝❡❧❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥❝♦❧✐❝ ❛♣♣r♦❛❝❤ ■❞❡❛ ✿ ❛❝❝❡❧❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥❝♦❧✐❝ ❛♣♣r♦❛❝❤ ■❞❡❛ ✿ ❛❝❝❡❧❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥❝♦❧✐❝ ❛♣♣r♦❛❝❤ ■❞❡❛ ✿ ❛❝❝❡❧❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥❝♦❧✐❝ ❛♣♣r♦❛❝❤ ■❞❡❛ ✿ ❛❝❝❡❧❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥❝♦❧✐❝ ❛♣♣r♦❛❝❤ ■❞❡❛ ✿ ❛❝❝❡❧❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❙♦♠❡ ❦♥♦✇♥ t♦♦❧s ♦❢ ❈❇❚ ✭♥♦♥ ❡①❤❛✉st✐✈❡ ✳✳✳✮ ❈ ✲ ❈✰✰ ✿ ❈✉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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❖✉t❧✐♥❡ ✶ ●❡♥❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧✐t✐❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧✐t✐❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧✐t✐❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧✐t✐❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ❛❜♦✉t ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣❄ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ✭▼❇❚✮ → t❡st✐♥❣ ✇✐t❤ t❤❡ ❛❜✐❧✐t② t♦ ❞❡t❡❝t ❢❛✉❧ts ✇❤✐❝❤ ❞♦ ♥♦t ❝♦♥❢♦r♠ t♦ ❛ ♠♦❞❡❧ ❝❛❧❧❡❞ s♣❡❝✐✜❝❛t✐♦♥✳ specifies Implementation Under Test conforms (IUT) Specification → ♣♦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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ❛❜♦✉t ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣❄ ✭✷✮ ❚❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ❛❜♦✉t ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣❄ ✭✷✮ ❚❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧ s❝❤❡♠❛ Property P S P ? Specification S I conf S ? Implementation I ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧ s❝❤❡♠❛ Property P S P ? VERIFICATION Specification S I conf S ? Implementation I ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧ s❝❤❡♠❛ Property P S P ? Specification S I conf S ? TEST Implementation I ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧ s❝❤❡♠❛ Property P S P ? Specification S I conf S ? observation ? Implementation I Test cases Verdict control ! ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧ s❝❤❡♠❛ Property P S P ? Specification S Test Generation I conf S ? observation ? Implementation I Test cases Verdict control ! ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ●❡♥❡r❛❧ s❝❤❡♠❛ Property P Test Purpose S P ? Specification S Test Generation I conf S ? observation ? Implementation I Test cases Verdict control ! ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼♦❞❡❧❧✐♥❣ t❤❡ s②st❡♠ ✭✶✮ ❋✐♥❞ ❛ ✏❣♦♦❞✑ ♠♦❞❡❧ ◆❡❝❡ss✐t② t♦ ✜♥❞ ❛ ❢♦r♠❛❧ ♠♦❞❡❧ ❛❞❛♣t❡❞ t♦ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ⇒ ♥♦t r❡❛❧✐st✐❝✱ ♥♦ ✐♥t❡r❡st ❚♦♦ ❛❜str❛❝t = ❚♦♦ ❞❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼♦❞❡❧❧✐♥❣ t❤❡ s②st❡♠ ✭✷✮ ❊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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼♦❞❡❧❧✐♥❣ t❤❡ s②st❡♠ ✭✷✮ ❊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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼♦❞❡❧❧✐♥❣ t❤❡ s②st❡♠ ✭✷✮ ❊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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼♦❞❡❧❧✐♥❣ t❤❡ s②st❡♠ ✭✷✮ ❊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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼♦❞❡❧❧✐♥❣ t❤❡ s②st❡♠ ✭✸✮ ❊①❛♠♣❧❡ ♦❢ ▼♦❞❡❧ ✿ ✭■❖✮▲❚❙ ✭▲❚❙ ✇✐t❤ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✮ ❆ ✭✈❡r② ✈❡r②✮ s✐♠♣❧✐✜❡❞ ❞✐❣✐❝♦❞❡✳ ✐♥♣✉ts ✭r❡s♣✳ ♦✉t♣✉ts✮ ❧❛❜❡❧❧❡❞ ✇✐t❤ ✏❄✑ ✭r❡s♣✳ ✏✦✑✮ s 0 ? reset ? digit s 1 τ 2 τ 1 ? reset s 2 ! beep ? digit s 7 s 3 ! open ! alarm τ 3 s 4 s 5 s 6 τ 4 P♦ss✐❜✐❧✐t② t♦ ❛❞❞ t✐♠❡ ✭❚■❖❚❙✱ ❚❆■❖✮✱ ❞❛t❛ ✈❛r✐❛❜❧❡s ✭■❖❙❚❙✮ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✻✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼♦❞❡❧❧✐♥❣ t❤❡ s②st❡♠ ✭✸✮ ❊①❛♠♣❧❡ ♦❢ ▼♦❞❡❧ ✿ ✭■❖✮▲❚❙ ✭▲❚❙ ✇✐t❤ ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✮ ❆ ✭✈❡r② ✈❡r②✮ s✐♠♣❧✐✜❡❞ ❞✐❣✐❝♦❞❡✳ ✐♥♣✉ts ✭r❡s♣✳ ♦✉t♣✉ts✮ ❧❛❜❡❧❧❡❞ ✇✐t❤ ✏❄✑ ✭r❡s♣✳ ✏✦✑✮ s 0 ? reset ? digit s 1 τ 2 τ 1 ? reset s 2 ! beep ? digit s 7 s 3 ! open ! alarm τ 3 s 4 s 5 s 6 τ 4 P♦ss✐❜✐❧✐t② t♦ ❛❞❞ t✐♠❡ ✭❚■❖❚❙✱ ❚❆■❖✮✱ ❞❛t❛ ✈❛r✐❛❜❧❡s ✭■❖❙❚❙✮ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✸✻✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❖✉t❧✐♥❡ ✶ ●❡♥❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❘❡❢❡r❡♥❝❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡ 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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡ 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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡ 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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ▼❡❛❧② ▼❛❝❤✐♥❡ 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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❈♦♥❢♦r♠❛♥❝❡ t❡st✐♥❣ Pr♦❜❧❡♠ ❞❡s❝r✐♣t✐♦♥ ❙♣❡❝✐✜❝❛t✐♦♥ M S ✱ ▼❡❛❧② ♠❛❝❤✐♥❡✱ ❦♥♦✇♥ ■♠♣❧❡♠❡♥t❛t✐♦♥ M I ▼❡❛❧② ♠❛❝❤✐♥❡✱ ✉♥❦♥♦✇♥✱ ♦♥❧② ✐♥♣✉ts✴♦✉t♣✉ts ❛r❡ ♦❜s❡r✈❛❜❧❡ ❆✐♠ ✿ ❯s❡ t❡st s❡q✉❡♥❝❡s t♦ ❝❤❡❝❦ ✐❢ M I ✐s ❡q✉✐✈❛❧❡♥t ❛ t♦ M S ✱ ✐✳❡✳ M I ❝♦♥❢♦r♠s t♦ M S ❛ ❍❡r❡ ✏❡q✉✐✈❛❧❡♥t✑ ♠❡❛♥s ✐s♦♠♦r♣❤✐❝ → ❣❡♥❡r❛❧❧②✱ M S ❛♥❞ M I s✉♣♣♦s❡❞ t♦ ❜❡ ♠✐♥✐♠❛❧ ❛♥❞ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ✭✉s✉❛❧❧② ❡①✐st❡♥❝❡ ♦❢ ❛ r❡s❡t ❛❝t✐♦♥✮ → M S ❛♥❞ M I ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ✭✉s✉❛❧❧② ♥♦t ♥❡❝❡ss❛r②✮ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✵✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st s②♥t❤❡s✐s ❋❛✉❧t ♠♦❞❡❧ ❈❤❡❝❦✐♥❣ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ M I ❛♥❞ M S ♠❡❛♥s ❝❤❡❝❦✐♥❣ ✐❢ M I ❤❛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✐♦♥✮ ●♦ t♦ s ❆♣♣❧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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st s②♥t❤❡s✐s ❋❛✉❧t ♠♦❞❡❧ ❈❤❡❝❦✐♥❣ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ M I ❛♥❞ M S ♠❡❛♥s ❝❤❡❝❦✐♥❣ ✐❢ M I ❤❛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✐♦♥✮ ●♦ t♦ s ❆♣♣❧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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ■❞❡♥t✐✜❝❛t✐♦♥ s❡q✉❡♥❝❡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, ∃ UIO s ∈ I ∗ , ∀ s ′ ∈ S \ { s } , λ ∗ ( s, UIO s ) � = λ ∗ ( s ′ , UIO s ) ✭❯■❖ ♠❡t❤♦❞ ❬❙❉✽✽❪✮ ❲ s❡t ♦❢ x ij s❡q✉❡♥❝❡s ✿ ∀ s i , s j ∈ S, ∃ x ij ∈ I ∗ , λ ∗ ( s i , x ij ) � = λ ∗ ( s j , x ij ) ✭❲ ♠❡t❤♦❞ ❬❈❤♦✼✽❪✮ → ❲ s❡t ❛❧✇❛②s ❡①✐sts ✐♥ ❝❛s❡ ♦❢ ♠✐♥✐♠❛❧ ♠❛❝❤✐♥❡✱ ♦t❤❡rs ♠❛② ♥♦t ❡①✐st ✷ ❯♥✐q✉❡ ■♥♣✉t ❖✉t♣✉t ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✷✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ■❞❡♥t✐✜❝❛t✐♦♥ s❡q✉❡♥❝❡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, ∃ UIO s ∈ I ∗ , ∀ s ′ ∈ S \ { s } , λ ∗ ( s, UIO s ) � = λ ∗ ( s ′ , UIO s ) ✭❯■❖ ♠❡t❤♦❞ ❬❙❉✽✽❪✮ ❲ s❡t ♦❢ x ij s❡q✉❡♥❝❡s ✿ ∀ s i , s j ∈ S, ∃ x ij ∈ I ∗ , λ ∗ ( s i , x ij ) � = λ ∗ ( s j , x ij ) ✭❲ ♠❡t❤♦❞ ❬❈❤♦✼✽❪✮ → ❲ s❡t ❛❧✇❛②s ❡①✐sts ✐♥ ❝❛s❡ ♦❢ ♠✐♥✐♠❛❧ ♠❛❝❤✐♥❡✱ ♦t❤❡rs ♠❛② ♥♦t ❡①✐st ✷ ❯♥✐q✉❡ ■♥♣✉t ❖✉t♣✉t ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✷✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ■❞❡♥t✐✜❝❛t✐♦♥ s❡q✉❡♥❝❡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, ∃ UIO s ∈ I ∗ , ∀ s ′ ∈ S \ { s } , λ ∗ ( s, UIO s ) � = λ ∗ ( s ′ , UIO s ) ✭❯■❖ ♠❡t❤♦❞ ❬❙❉✽✽❪✮ ❲ s❡t ♦❢ x ij s❡q✉❡♥❝❡s ✿ ∀ s i , s j ∈ S, ∃ x ij ∈ I ∗ , λ ∗ ( s i , x ij ) � = λ ∗ ( s j , x ij ) ✭❲ ♠❡t❤♦❞ ❬❈❤♦✼✽❪✮ → ❲ s❡t ❛❧✇❛②s ❡①✐sts ✐♥ ❝❛s❡ ♦❢ ♠✐♥✐♠❛❧ ♠❛❝❤✐♥❡✱ ♦t❤❡rs ♠❛② ♥♦t ❡①✐st ✷ ❯♥✐q✉❡ ■♥♣✉t ❖✉t♣✉t ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✷✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ■❞❡♥t✐✜❝❛t✐♦♥ s❡q✉❡♥❝❡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, ∃ UIO s ∈ I ∗ , ∀ s ′ ∈ S \ { s } , λ ∗ ( s, UIO s ) � = λ ∗ ( s ′ , UIO s ) ✭❯■❖ ♠❡t❤♦❞ ❬❙❉✽✽❪✮ ❲ s❡t ♦❢ x ij s❡q✉❡♥❝❡s ✿ ∀ s i , s j ∈ S, ∃ x ij ∈ I ∗ , λ ∗ ( s i , x ij ) � = λ ∗ ( s j , x ij ) ✭❲ ♠❡t❤♦❞ ❬❈❤♦✼✽❪✮ → ❲ s❡t ❛❧✇❛②s ❡①✐sts ✐♥ ❝❛s❡ ♦❢ ♠✐♥✐♠❛❧ ♠❛❝❤✐♥❡✱ ♦t❤❡rs ♠❛② ♥♦t ❡①✐st ✷ ❯♥✐q✉❡ ■♥♣✉t ❖✉t♣✉t ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✷✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st ❣❡♥❡r❛t✐♦♥ ✭✶✮ ❆✐♠ ✿ 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❡ UIO sj i/o ❈♦♥s✐❞❡r✐♥❣ t❤❡ tr❛♥s✐t✐♦♥s s i → s j → s k ✱ ✜♥❞ ❛ ♠✐♥✐♠❛❧ ❝✐r❝✉✐t r✉♥♥✐♥❣ ♦♥❝❡ t❤r♦✉❣❤ t❤❡s❡ tr❛♥s✐t✐♦♥s → ❘✉r❛❧ ❈❤✐♥❡s❡ P♦st♠❛♥ ♣r♦❜❧❡♠ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st ❣❡♥❡r❛t✐♦♥ ✭✶✮ ❆✐♠ ✿ 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❡ UIO sj i/o ❈♦♥s✐❞❡r✐♥❣ t❤❡ tr❛♥s✐t✐♦♥s s i → s j → s k ✱ ✜♥❞ ❛ ♠✐♥✐♠❛❧ ❝✐r❝✉✐t r✉♥♥✐♥❣ ♦♥❝❡ t❤r♦✉❣❤ t❤❡s❡ tr❛♥s✐t✐♦♥s → ❘✉r❛❧ ❈❤✐♥❡s❡ P♦st♠❛♥ ♣r♦❜❧❡♠ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✸✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st ❣❡♥❡r❛t✐♦♥ ✭✷✮ ❉❙ ❬●♦♥✼✵❪ ❈♦✉❧❞ ❜❡ 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✳ ∀ s i , s j ∈ S, ∃ x ij ∈ W, λ ∗ ( s i , x ij ) � = λ ∗ ( s j , x ij ) ◆♦t✐♥❣ X · Y t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ X ✇✐t❤ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ Y ✱ ❣❡♥❡r❛t❡ { reset } · P · W W p ✱ UIO p ✱ UIO v ✱ DS ✇✐t❤♦✉t r❡s❡ts✱ ❆❞❛♣t❛t✐✈❡ DS ✱ HSI ✱ ✳✳✳ P♦ss✐❜✐❧✐t② t♦ ❜❡ ♠♦r❡ ❡✣❝✐❡♥t ❜② ✉s✐♥❣ ❆❞❛♣t❛t✐✈❡ ❙❡q✉❡♥❝❡s ✸ ❝❛s❡ ✇❤❡r❡ M I ❛♥❞ M S ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✹✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st ❣❡♥❡r❛t✐♦♥ ✭✷✮ ❉❙ ❬●♦♥✼✵❪ ❈♦✉❧❞ ❜❡ 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✳ ∀ s i , s j ∈ S, ∃ x ij ∈ W, λ ∗ ( s i , x ij ) � = λ ∗ ( s j , x ij ) ◆♦t✐♥❣ X · Y t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ X ✇✐t❤ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ Y ✱ ❣❡♥❡r❛t❡ { reset } · P · W W p ✱ UIO p ✱ UIO v ✱ DS ✇✐t❤♦✉t r❡s❡ts✱ ❆❞❛♣t❛t✐✈❡ DS ✱ HSI ✱ ✳✳✳ P♦ss✐❜✐❧✐t② t♦ ❜❡ ♠♦r❡ ❡✣❝✐❡♥t ❜② ✉s✐♥❣ ❆❞❛♣t❛t✐✈❡ ❙❡q✉❡♥❝❡s ✸ ❝❛s❡ ✇❤❡r❡ M I ❛♥❞ M S ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✹✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st ❣❡♥❡r❛t✐♦♥ ✭✷✮ ❉❙ ❬●♦♥✼✵❪ ❈♦✉❧❞ ❜❡ 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✳ ∀ s i , s j ∈ S, ∃ x ij ∈ W, λ ∗ ( s i , x ij ) � = λ ∗ ( s j , x ij ) ◆♦t✐♥❣ X · Y t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ X ✇✐t❤ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ Y ✱ ❣❡♥❡r❛t❡ { reset } · P · W W p ✱ UIO p ✱ UIO v ✱ DS ✇✐t❤♦✉t r❡s❡ts✱ ❆❞❛♣t❛t✐✈❡ DS ✱ HSI ✱ ✳✳✳ P♦ss✐❜✐❧✐t② t♦ ❜❡ ♠♦r❡ ❡✣❝✐❡♥t ❜② ✉s✐♥❣ ❆❞❛♣t❛t✐✈❡ ❙❡q✉❡♥❝❡s ✸ ❝❛s❡ ✇❤❡r❡ M I ❛♥❞ M S ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✹✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st ❣❡♥❡r❛t✐♦♥ ✭✷✮ ❉❙ ❬●♦♥✼✵❪ ❈♦✉❧❞ ❜❡ 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✳ ∀ s i , s j ∈ S, ∃ x ij ∈ W, λ ∗ ( s i , x ij ) � = λ ∗ ( s j , x ij ) ◆♦t✐♥❣ X · Y t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ X ✇✐t❤ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ Y ✱ ❣❡♥❡r❛t❡ { reset } · P · W W p ✱ UIO p ✱ UIO v ✱ DS ✇✐t❤♦✉t r❡s❡ts✱ ❆❞❛♣t❛t✐✈❡ DS ✱ HSI ✱ ✳✳✳ P♦ss✐❜✐❧✐t② t♦ ❜❡ ♠♦r❡ ❡✣❝✐❡♥t ❜② ✉s✐♥❣ ❆❞❛♣t❛t✐✈❡ ❙❡q✉❡♥❝❡s ✸ ❝❛s❡ ✇❤❡r❡ M I ❛♥❞ M S ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ st❛t❡s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✹✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❖✉t❧✐♥❡ ✶ ●❡♥❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❘❡❢❡r❡♥❝❡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❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ■♥♣✉t ❖✉t♣✉t ▲❛❜❡❧❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠ ✭■❖▲❚❙✮ M = ( Q M , A M , − → M , q M 0 ) ✇✐t❤ ✿ Q M s❡t ♦❢ st❛t❡s s 0 ? reset 0 ∈ Q M ✐♥✐t✐❛❧ st❛t❡ q M ? digit A M ❛❝t✐♦♥ ❛❧♣❤❛❜❡t✱ s 1 τ 2 τ 1 ? reset A M ✐♥♣✉t ❛❧♣❤❛❜❡t ✭✇✐t❤ ? ✮ I s 2 A M O ♦✉t♣✉t ❛❧♣❤❛❜❡t ✭✇✐t❤ ! ✮ ! beep I M ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ✭ τ k ✮ ? digit s 7 → M ⊆ Q M × A M × Q M s 3 − ! open ! alarm τ 3 tr❛♥s✐t✐♦♥ r❡❧❛t✐♦♥ s 4 s 5 s 6 τ 4 A M V IS = A M I ∪ A M O s❡t ♦❢ ✈✐s✐❜❧❡ ❛❝t✐♦♥s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✼✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ■♥♣✉t ❖✉t♣✉t ▲❛❜❡❧❧❡❞ ❚r❛♥s✐t✐♦♥ ❙②st❡♠ ✭■❖▲❚❙✮ M = ( Q, A, − → , q 0 ) ✇✐t❤ ✿ Q s❡t ♦❢ st❛t❡s s 0 ? r ? d q 0 ∈ Q ✐♥✐t✐❛❧ st❛t❡ s 1 A ❛❝t✐♦♥ ❛❧♣❤❛❜❡t✱ τ 1 τ 2 ? r A I ✐♥♣✉t ❛❧♣❤❛❜❡t ✭✇✐t❤ ? ✮ s 2 A O ♦✉t♣✉t ❛❧♣❤❛❜❡t ✭✇✐t❤ ! ✮ ! b I ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ✭ τ k ✮ ? d s 7 s 3 − →⊆ Q × A × Q ! o ! a τ 3 tr❛♥s✐t✐♦♥ r❡❧❛t✐♦♥ s 4 s 5 s 6 τ 4 A V IS = A I ∪ A O s❡t ♦❢ ✈✐s✐❜❧❡ ❛❝t✐♦♥s ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✼✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❘✉♥s ✴ ❚r❛❝❡s ❘✉♥s✿ ❛❧t❡r♥❛t❡ s❡q✉❡♥❝❡s ♦❢ st❛t❡s ❛♥❞ ❛❝t✐♦♥s ✜r❡❛❜❧❡ ❜t✇ t❤♦s❡ st❛t❡s ? d τ 1 ? d ! o s 0 → s 1 → s 2 → s 3 → s 4 ∈ Runs ( M ) s 0 ? r ? d Traces ✿ ♣r♦❥❡❝t✐♦♥s ♦❢ Runs s 1 ♦♥ ✈✐s✐❜❧❡ ❛❝t✐♦♥s✿ τ 1 ? r τ 2 Traces ( M ) = { ε, ? d, ? r, ? d. ? r, ? r. ? d, ? d. ! b, ... } s 2 ! b ? d s 7 P ❛❢t❡r σ ✿ s❡t ♦❢ st❛t❡s r❡❛❝❤❛❜❧❡ ❢r♦♠ P s 3 ❛❢t❡r ♦❜s❡r✈❛t✐♦♥ σ ✿ ! o ! a τ 3 { s 2 } ❛❢t❡r ? d. ! o = { s 0 , s 4 } s 4 s 5 s 6 τ 4 { s 0 } ❛❢t❡r ? d, ! a = ∅ M ❛❢t❡r σ � { q 0 } ❛❢t❡r σ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✽✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ◆♦♥✲❞❡t❡r♠✐♥✐s♠ M ✐s ❞❡t❡r♠✐♥✐st✐❝ ✐❢ ✐t ❤❛s ♥♦ ✐♥t❡r♥❛❧ ❛❝t✐♦♥✱ ❛♥❞ ∀ q, q ′ , q ′′ ∈ Q, ∀ a ∈ A V IS , ( q a → q ′ ∧ q → q ′′ ) ⇒ q ′ = q ′′ a − − τ s 2 ! a s 1 s 2 ? x ! b s 1 ! a s 3 s 4 s 3 ! a ◆♦t t♦ ❜❡ ❝♦♥❢✉s❡❞ ✇✐t❤ ✉♥❝♦♥tr♦❧❧❡❞ ❝❤♦✐❝❡ s 2 ! a ? x s 1 s 3 ! b ❉❡t❡r♠✐♥✐③❛t✐♦♥✿ det ( M ) = (2 Q , A V IS , − → det , q 0 ❛❢t❡r ǫ ) ✇✐t❤ → det P ′ ⇔ P, P ′ ∈ 2 Q , a ∈ A V IS ❛♥❞ P ′ = P ❛❢t❡r a ✳ a − P Traces ( M ) = Traces ( det ( M )) ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✾✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ◆♦♥✲❞❡t❡r♠✐♥✐s♠ M ✐s ❞❡t❡r♠✐♥✐st✐❝ ✐❢ ✐t ❤❛s ♥♦ ✐♥t❡r♥❛❧ ❛❝t✐♦♥✱ ❛♥❞ ∀ q, q ′ , q ′′ ∈ Q, ∀ a ∈ A V IS , ( q a → q ′ ∧ q → q ′′ ) ⇒ q ′ = q ′′ a − − τ s 2 ! a s 1 s 2 ? x ! b s 1 ! a s 3 s 4 s 3 ! a ◆♦t t♦ ❜❡ ❝♦♥❢✉s❡❞ ✇✐t❤ ✉♥❝♦♥tr♦❧❧❡❞ ❝❤♦✐❝❡ s 2 ! a ? x s 1 s 3 ! b ❉❡t❡r♠✐♥✐③❛t✐♦♥✿ det ( M ) = (2 Q , A V IS , − → det , q 0 ❛❢t❡r ǫ ) ✇✐t❤ → det P ′ ⇔ P, P ′ ∈ 2 Q , a ∈ A V IS ❛♥❞ P ′ = P ❛❢t❡r a ✳ a − P Traces ( M ) = Traces ( det ( M )) ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✹✾✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❖❜s❡r✈❛t✐♦♥ ♦❢ q✉✐❡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❡❞✳ s 0 ? r a ◆♦t❛t✐♦♥ ✿ Γ( q ) � { a ∈ A | q ? d →} s 1 ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ τ 1 ? r Γ( q ) = ∅ ✳ τ 2 s 2 ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠s ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ ! b Γ( q ) ⊆ A I ✳ ? d s 7 ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ s 3 ! o ! a τ 3 ∃ τ 1 , ...τ n : q τ 1 ...τ n − → q ✳ s 4 s 5 s 6 τ 4 quiescent ( M ) = deadlock ( M ) ∪ livelock ( M ) ∪ outputlock ( M ) ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❖❜s❡r✈❛t✐♦♥ ♦❢ q✉✐❡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❡❞✳ s 0 ? r a ◆♦t❛t✐♦♥ ✿ Γ( q ) � { a ∈ A | q ? d →} s 1 ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ τ 1 ? r Γ( q ) = ∅ ✳ τ 2 s 2 ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠s ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ ! b Γ( q ) ⊆ A I ✳ ? d s 7 ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ s 3 ! o ! a τ 3 ∃ τ 1 , ...τ n : q τ 1 ...τ n − → q ✳ s 4 s 5 s 6 τ 4 quiescent ( M ) = deadlock ( M ) ∪ livelock ( M ) ∪ outputlock ( M ) ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❖❜s❡r✈❛t✐♦♥ ♦❢ q✉✐❡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❡❞✳ s 0 ? r a ◆♦t❛t✐♦♥ ✿ Γ( q ) � { a ∈ A | q ? d →} s 1 ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ τ 1 ? r Γ( q ) = ∅ ✳ τ 2 s 2 ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠ ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ ! b Γ( q ) ⊆ A I ✳ ? d s 7 ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ s 3 ! o ! a τ 3 ∃ τ 1 , ...τ n : q τ 1 ...τ n − → q ✳ s 4 s 5 s 6 τ 4 quiescent ( M ) = deadlock ( M ) ∪ livelock ( M ) ∪ outputlock ( M ) ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❖❜s❡r✈❛t✐♦♥ ♦❢ q✉✐❡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❡❞✳ s 0 ? r a ◆♦t❛t✐♦♥ ✿ Γ( q ) � { a ∈ A | q ? d →} s 1 ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ τ 1 ? r Γ( q ) = ∅ ✳ τ 2 s 2 ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠ ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ ! b Γ( q ) ⊆ A I ✳ ? d s 7 ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ s 3 ! o ! a τ 3 ∃ τ 1 , ...τ n : q τ 1 ...τ n − → q ✳ s 4 s 5 s 6 τ 4 quiescent ( M ) = deadlock ( M ) ∪ livelock ( M ) ∪ outputlock ( M ) ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❖❜s❡r✈❛t✐♦♥ ♦❢ q✉✐❡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❡❞✳ s 0 ? r a ◆♦t❛t✐♦♥ ✿ Γ( q ) � { a ∈ A | q ? d →} s 1 ❞❡❛❞❧♦❝❦ ✿ ♥♦ ♣♦ss✐❜❧❡ ❡✈♦❧✉t✐♦♥ ✿ τ 1 ? r Γ( q ) = ∅ ✳ τ 2 s 2 ♦✉t♣✉t❧♦❝❦ ✿ s②st❡♠ ✇❛✐t✐♥❣ ❢♦r ❛♥ ❛❝t✐♦♥ ✿ ! b Γ( q ) ⊆ A I ✳ ? d s 7 ❧✐✈❡❧♦❝❦ ✿ ✐♥t❡r♥❛❧ ❛❝t✐♦♥s ❧♦♦♣ ✿ s 3 ! o ! a τ 3 ∃ τ 1 , ...τ n : q τ 1 ...τ n − → q ✳ s 4 s 5 s 6 τ 4 quiescent ( M ) = deadlock ( M ) ∪ livelock ( M ) ∪ outputlock ( M ) ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✵✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❙✉s♣❡♥s✐♦♥ ❛✉t♦♠❛t♦♥ ◗✉✐❡s❝❡♥❝❡ ✿ s♣❡❝✐❛❧ ♦✉t♣✉t δ ! δ s 0 ? r ❙✉s♣❡♥s✐♦♥ ❛✉t♦♠❛t♦♥ ∆( M ) ? d ∆( M ) ❂ ❙♣❡❝✐✜❝❛t✐♦♥ M ✰ δ ✲tr❛♥s✐t✐♦♥s s 1 ♦♥ q✉✐❡s❝❡♥t st❛t❡s τ 1 ? r τ 2 s 2 ! b ? d s 7 ! δ ❙✉s♣❡♥s✐♦♥ tr❛❝❡s s 3 ! o ! a τ 3 STraces ( M ) � Traces (∆( M )) = s 4 s 5 s 6 Traces ( det (∆( M ))) τ 4 ! δ ! δ ❆✳ ❘♦❧❧❡t ✲ ❚❆❘❖❚✷✵✶✻ ✲ P❛r✐s ✭❋r❛♥❝❡✮ ✲ ❏✉❧② ✷✵✶✻ ✺✶✴✼✸

●❡♥❡r❛❧✐t✐❡s ♦♥ t❡st✐♥❣ ❙♦✉r❝❡ ❈♦❞❡ ❇❛s❡❞ ❚❡st✐♥❣ ✭❙❈❇❚✮ ❋✉♥❝t✐♦♥❛❧ t❡st✐♥❣ ✲ ▼♦❞❡❧ ❇❛s❡❞ ❚❡st✐♥❣ ❈♦♥❝❧✉s✐♦♥ ❚❡st✐♥❣ ❢r❛♠❡✇♦r❦ ❙♣❡❝✐✜❝❛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 ✷✵✶✻ ✺✷✴✼✸