❈✉rs♦ ❞❡ ❙✐st❡♠❛s ❞❡ ■♥❢♦r♠❛çã♦ ✴ ❊♥❣❡♥❤❛r✐❛ ❞❡ ❈♦♠♣✉t❛çã♦ ❉❊❈❊❆ ✲ ❯❋❖P
▲✐♥❣✉❛❣❡♥s ❞❡ Pr♦❣r❛♠❛çã♦ ✶♦ s❡♠❡str❡ ❞❡ ✷✵✶✶
Pr♦❢❡ss♦r✿ ❘♦❞r✐❣♦ ●❡r❛❧❞♦ ❘✐❜❡✐r♦
❡✲♠❛✐❧✿ r♦❞r✐❣♦❣r✐❜❡✐r♦❅❞❡❝❡❛✳✉❢♦♣✳❜r
✶ ❯♠ ■♥t❡r♣r❡t❛❞♦r ♣❛r❛ ❛ ▲✐♥❣✉❛❣❡♠ ■▼P
✶✳✶ ■♥tr♦❞✉çã♦
◆❡st❡ tr❛❜❛❧❤♦ ✈♦❝ê ✐rá ✐♠♣❧❡♠❡♥t❛r ✉♠ ✐♥t❡r♣r❡t❛❞♦r ❞❡ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ✐♠♣❡r❛t✐✈❛ s✐♠♣❧❡s ❞❡♥♦♠✐♥❛❞❛ ■▼P✳ P❛r❛ ✐st♦✱ s❡rá ♥❡❝❡ssár✐♦ ✐♠♣❧❡♠❡♥t❛r ❛s ❞✐✈❡rs❛s ♣❛rt❡s q✉❡ ❝♦♠♣õ❡ ♦ ✐♥t❡r♣r❡t❛❞♦r✿ ♦ ❛♥❛❧✐s❛❞♦r ❧é①✐❝♦✱ s✐♥tát✐❝♦✱ s❡♠â♥t✐❝♦ ❡ ♦ ♠❡❝❛♥✐s♠♦ ❞❡ ❡①❡❝✉çã♦✳
❆s ♣ró①✐♠❛s s❡çõ❡s ❛♣r❡s❡♥t❛♠ ❡♠ ❞❡t❛❧❤❡s ❛ ❧✐♥❣✉❛❣❡♠ ■▼P ❞❡s❝r❡✲ ✈❡♥❞♦ s❡✉s ❝♦♠♣♦♥❡♥t❡s ❧é①✐❝♦s✱ s✉❛ ❣r❛♠át✐❝❛✱ s❡✉ ♠❡❝❛♥✐s♠♦ ❞❡ ✈❡r✐✜✲ ❝❛çã♦ ❞❡ t✐♣♦s ❡ s✉❛ s❡♠â♥t✐❝❛ ♦♣❡r❛❝✐♦♥❛❧✳ ◆❛ s❡q✉ê♥❝✐❛ é ❛♣r❡s❡♥t❛❞❛ ❛ ❡str✉t✉r❛ ❞♦ ❝ó❞✐❣♦ ❢♦r♥❡❝✐❞♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❡ ❞❡✈❡ s❡r ✐♠♣❧❡♠❡♥t❛❞♦✳
✷ ❯♠❛ ■♥tr♦❞✉çã♦ ❛ ▲✐♥❣✉❛❣❡♠ ■▼P
Pr♦❣r❛♠❛s ■▼P sã♦ ❡①tr❡♠❛♠❡♥t❡ s✐♠♣❧❡s ❡ ❝♦♥s✐st❡♠ ❛♣❡♥❛s ❞❡ ✉♠❛ ❧✐st❛ ❞❡ ❝♦♠❛♥❞♦s q✉❡ sã♦ ❡①❡❝✉t❛❞♦s s❡q✉ê♥❝✐❛❧♠❡♥t❡✳ ❈❛❞❛ ❝♦♠❛♥❞♦ ♣♦❞❡ s❡r ❝❧❛ss✐✜❝❛❞♦ ❝♦♠♦✿
• ❈♦♠❛♥❞♦s ❞❡ ❡♥tr❛❞❛ ✴ s❛í❞❛✿ ❊①✐st❡♠ ❛♣❡♥❛s ❝♦♠❛♥❞♦s ♣❛r❛ ❧❡✐t✉r❛ ❞❡ ✈❛❧♦r❡s ♥♦ t❡❝❧❛❞♦ ✭❝♦♠❛♥❞♦ r❡❛❞✮ ❡ ✉♠ ❝♦♠❛♥❞♦ ♣❛r❛ ❛ ✐♠♣r❡ssã♦ ❞❡ r❡s✉❧t❛❞♦s ✭❝♦♠❛♥❞♦ ♣r✐♥t✮✳ ❖ ❝♦♠❛♥❞♦ r❡❛❞ r❡❝❡❜❡ ❝♦♠♦ ♣❛râ♠❡✲ tr♦ ✉♠❛ ✈❛r✐á✈❡❧ ❡ ❛ss♦❝✐❛ ❛ ❡st❛ ♦ ✈❛❧♦r ❧✐❞♦ ❛ ♣❛rt✐r ❞♦ t❡❝❧❛❞♦✳ P♦r s✉❛ ✈❡③✱ ♦ ❝♦♠❛♥❞♦ ♣r✐♥t ♣♦❞❡ r❡❝❡❜❡r ❡①♣r❡ssõ❡s ❛r❜✐trár✐❛s ❝♦♠♦ ♣❛râ♠❡tr♦✳
• ❆tr✐❜✉✐çã♦ ❞❡ ✈❛❧♦r❡s✿ ❆tr✐❜✉✐çã♦ ❞❡ ✈❛❧♦r❡s ♣♦ss✉✐ ♦ s✐❣♥✐✜❝❛❞♦ ✉s✉❛❧ ❞❡ ❧✐♥❣✉❛❣❡♥s ✐♠♣❡r❛t✐✈❛s✱ ♣❡r♠✐t✐♥❞♦ ❛ss♦❝✐❛r ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❡①♣r❡s✲ sã♦ ❡ ❛ ✉♠❛ ✈❛r✐á✈❡❧ ✈✳
• ❈♦♠❛♥❞♦ ❝♦♥❞✐❝✐♦♥❛❧✿ ❈♦♠❛♥❞♦s ❞❛ ❢♦r♠❛ ✐❢ ❡ t❤❡♥ ❝✶ ❡❧s❡ ❝✷✱ ♦♥❞❡ ❡ é ✉♠❛ ❡①♣r❡ssã♦ ❜♦♦❧❡❛♥❛ ❡ ❝✶✱ ❝✷ sã♦ ❜❧♦❝♦s ❞❡ ❝♦♠❛♥❞♦s✳ ❇❧♦❝♦s sã♦ ❞❡❧✐♠✐t❛❞♦s ♣♦r ✏④✑ ❡ ✏⑥✑✳
• ▲♦♦♣✿ ❈♦♠❛♥❞♦s ❞❛ ❢♦r♠❛ ✇❤✐❧❡ ❡ ❞♦ ❝✱ ♦♥❞❡ ❡ é ✉♠❛ ❡①♣r❡ssã♦ ❜♦♦❧❡❛♥❛ ❡ ❝ é ✉♠ ❜❧♦❝♦ ❞❡ ❝♦♠❛♥❞♦s✳
• ❈♦♠❛♥❞♦ s❦✐♣✿ ❈♦♠❛♥❞♦ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ♦♣❡r❛çã♦ q✉❡ ♥ã♦ r❡❛❧✐③❛ ♥❡♥❤✉♠❛ ❛✈❛❧✐❛çã♦✱ ♥❡♠ ❡❢❡✐t♦ ❝♦❧❛t❡r❛❧✳ ➱ ✉♠ ❝♦♠❛♥❞♦ q✉❡ ♥ã♦ ✏❢❛③ ♥❛❞❛✑✳
❆♥t❡s ❞❡ s❡ ❛♣r❡s❡♥t❛r ❛ ❡s♣❡❝✐✜❝❛çã♦ ❢♦r♠❛❧ ❞❛ ❧✐♥❣✉❛❣❡♠ ■▼P✱ é ❛♣r❡✲ s❡♥t❛❞♦ ✉♠ ♣r♦❣r❛♠❛ ❞❡ ❡①❡♠♣❧♦✳ ❊st❡ ♣r♦❣r❛♠❛ r❡❛❧✐③❛ ♦ ❝á❧❝✉❧♦ ❞♦ ❢❛t♦r✐❛❧ ❞❡ ♥ú♠❡r♦ ❧✐❞♦ ❛ ♣❛rt✐r ❞♦ t❡❝❧❛❞♦ ❡ ❛♦ ✜♥❛❧ ✐♠♣r✐♠❡ ♦ r❡s✉❧t❛❞♦ ❞♦ ❢❛t♦r✐❛❧✳ ✈❛r ④
✐♥t ①❀
✐♥t ❢❛t♦r✐❛❧❀ ⑥
♣r♦❣r❛♠ ④
♣r✐♥t ✧❞✐❣✐t❡ ✉♠ ♥✉♠❡r♦✿✧❀ r❡❛❞ ①❀
❢❛t♦r✐❛❧ ✿❂ ✶❀
✇❤✐❧❡ ✦ ✭① ❁❂ ✶✮ ❞♦ ④ ❢❛t♦r✐❛❧ ✿❂ ❢❛t♦r✐❛❧ ✯ ①❀ ① ✿❂ ① ✲ ✶❀
⑥
♣r✐♥t ❢❛t♦r✐❛❧❀ ⑥
❖ s❡♠â♥t✐❝❛ ❞♦ tr❡❝❤♦ ❞❡ ❝ó❞✐❣♦ ❛♥t❡r✐♦r ❞❡✈❡ s❡r ✐♥t✉✐t✐✈❛ ♣❛r❛ ♦ ❧❡✐t♦r✳ ❖ ❜❧♦❝♦ ✈❛r s❡r✈❡ ♣❛r❛ ❞❡✜♥✐r ❛s ✈❛r✐á✈❡✐s ❛ s❡r❡♠ ✉t✐❧✐③❛❞❛s ♥♦ ♣r♦❣r❛♠❛ ❡ ♦ ❜❧♦❝♦ ♣r♦❣r❛♠ ❝♦♥s✐st❡ ❞❛ ❞❡✜♥✐çã♦ ❞♦ ♣r♦❣r❛♠❛ ❡♠ s✐✳
❖ ♣r♦❝❡ss♦ ❞❡ ❡①❡❝✉çã♦ ❞❡ ✉♠ ♣r♦❣r❛♠❛ ■▼P é ❞✐✈✐❞♦ ♥❛s s❡❣✉✐♥t❡s ❡t❛♣❛s✿
• ❆♥á❧✐s❡ ❧é①✐❝❛✿ ❡st❛ ❡t❛♣❛ é r❡s♣♦♥sá✈❡❧ ♣♦r ❝♦♥✈❡rt❡r ❛ str✐♥❣ ❞❡ ❡♥✲ tr❛❞❛ ❡♠ ✉♠❛ ❧✐st❛ ❞❡ ❡❧❡♠❡♥t♦s r❡♣r❡s❡♥t❛t✐✈♦s ❞❛ s✐♥t❛①❡ ❞❛ ❧✐♥❣✉❛✲ ❣❡♠✳ ❉á✲s❡ ♦ ♥♦♠❡ ❞❡ t♦❦❡♥ ❛ ❝♦♠♣♦♥❡♥t❡s ✐♥❞✐✈✐sí✈❡✐s ❞❛ s✐♥t❛①❡ ❞❡ ✉♠❛ ❧✐♥❣✉❛❣❡♠✳ ❆ ❡s♣❡❝✐✜❝❛çã♦ ❞♦s t♦❦❡♥s ❞❡ ■▼P é ❛♣r❡s❡♥t❛❞❛ ♥❛ s❡çã♦ ✸✳✶✳
• ❆♥á❧✐s❡ s✐♥tát✐❝❛✿ ❡st❛ ❡t❛♣❛ é r❡s♣♦♥sá✈❡❧ ♣♦r ✈❡r✐✜❝❛r s❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ t♦❦❡♥s ♣r♦❞✉③✐❞❛ ♣❡❧♦ ❛♥❛❧✐s❛❞♦r ❧é①✐❝♦ ❡stá ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❣r❛✲ ♠át✐❝❛ ❞❛ ❧✐♥❣✉❛❣❡♠✳ ❖ r❡s✉❧t❛❞♦ ❞❛ ❛♥á❧✐s❡ s✐♥tát✐❝❛ é ✉♠❛ ❡str✉t✉r❛ ❞❡ ❞❛❞♦s ❞❡♥♦♠✐♥❛❞❛ ár✈♦r❡ ❞❡ s✐♥t❛①❡ ❛❜str❛t❛✳ ❆ ❡s♣❡❝✐✜❝❛çã♦ ❞❛ ❣r❛♠át✐❝❛ ❞❡ ■▼P ❡ ❞❛ ár✈♦r❡ ❞❡ s✐♥t❛①❡ ❛❜str❛t❛ sã♦ ❛♣r❡s❡♥t❛❞❛s r❡s♣❡❝t✐✈❛♠❡♥t❡ ♥❛s s❡çõ❡s ✸✳✷ ❡ ✹✳✷✳
• ❆♥á❧✐s❡ s❡♠â♥t✐❝❛✿ ❆ ❛♥á❧✐s❡ s❡♠â♥t✐❝❛ ♣r❡♦❝✉♣❛✲s❡ ❝♦♠ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡ ♥ã♦ ♣♦❞❡♠ s❡r ❞❡s❝r✐t❛s ♣♦r ❣r❛♠át✐❝❛s ❧✐✈r❡s ❞❡ ❝♦♥t❡①t♦✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ t❛❧ ♣r♦♣r✐❡❞❛❞❡ é ❛ ❝♦rr❡t✉❞❡ ❝♦♠ r❡s♣❡✐t♦ ❛♦ s✐st❡♠❛ ❞❡ t✐♣♦s ❞❡ ■▼P✳ ▼❛✐♦r❡s ❞❡t❛❧❤❡s s♦❜r❡ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ t✐♣♦s ❡♠ ■▼P é ❛♣r❡s❡♥t❛❞❛ ♥❛ s❡çã♦ ✸✳✸✳
• ■♥t❡r♣r❡t❛❞♦r✿ ❆♣ós t♦❞❛ ❛ ❛♥á❧✐s❡ s♦❜r❡ ♦ ♣r♦❣r❛♠❛✱ ❝❛s♦ ♥ã♦ s❡❥❛ ❡♥❝♦♥tr❛❞♦ ♥❡♥❤✉♠ ❡rr♦✱ ❡st❡ é ❡①❡❝✉t❛❞♦✳ ❆ ❡s♣❡❝✐✜❝❛çã♦ ❞❡ ❝♦♠♦ ♦ ♣r♦❣r❛♠❛ ❞❡✈❡ s❡r ❡①❡❝✉t❛❞♦ é ❞❛❞❛ ♥❛ s❡çã♦ ✸✳✹✳
✸ ❊s♣❡❝✐✜❝❛çã♦ ❞❛ ▲✐♥❣✉❛❣❡♠ ■▼P
◆❡st❛ s❡çã♦ sã♦ ❛♣r❡s❡♥t❛❞♦s ❝❛❞❛ ✉♠ ❞♦s ❝♦♠♣♦♥❡♥t❡s ❞❛ ❡s♣❡❝✐✜❝❛çã♦ ❢♦r♠❛❧ ❞❛ ❧✐♥❣✉❛❣❡♠ ■▼P✳
✸✳✶ ❊s♣❡❝✐✜❝❛çã♦ ▲é①✐❝❛ ❞❛ ▲✐♥❣✉❛❣❡♠ ■▼P
❖s t♦❦❡♥s ❞❛ ❧✐♥❣✉❛❣❡♠ ■▼P sã♦✱ ❡♠ s✉❛ ♠❛✐♦r✐❛✱ ♦♣❡r❛❞♦r❡s ❡ ♣❛❧❛✈r❛s r❡s❡r✈❛❞❛s ❞❛ ❧✐♥❣✉❛❣❡♠✳ ◆❛ t❛❜❡❧❛ s❡❣✉✐♥t❡✱ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s t♦❦❡♥s ❞❛ ❧✐♥❣✉❛❣❡♠✳ ❖ t✐♣♦ ❞❡ ❞❛❞♦s ❍❛s❦❡❧❧ q✉❡ ❞❡s❝r❡✈❡ ❡st❡s t♦❦❡♥s é ❛♣r❡s❡♥t❛❞♦ ♥❛ s❡çã♦ ✹✳✷✳✶✳
❉❡s❝r✐çã♦ ❚♦❦❡♥ ❉❡s❝r✐çã♦ ❚♦❦❡♥
❖♣❡r❛❞♦r ❞❡ s♦♠❛ ✰ ❖♣❡r❛❞♦r ❞❡ s✉❜tr❛çã♦ ✲ ❖♣❡r❛❞♦r ❞❡ ▼✉❧t✳ ✯ ❖♣❡r❛❞♦r ❞❡ ❞✐✈✐sã♦ ✴ ❖♣❡r❛❞♦r ❞❡ ✐❣✉❛❧❞❛❞❡ ❂ ❖♣❡r❛❞♦r ❞❡ ❝♦♠♣✳ ❁❂
◆❡❣❛çã♦ ! ❈♦♥❥✉♥çã♦ ✫
❉✐s❥✉♥çã♦ | ❋✐♠ ❞❡ ❝♦♠❛♥❞♦ ❀
❖♣❡r❛❞♦r ❆tr✐❜✉✐çã♦ ✿❂ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ✐❢ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ t❤❡♥ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ❡❧s❡ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ✇❤✐❧❡ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ❞♦ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ r❡❛❞ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ s❦✐♣ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ♣r✐♥t P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ✈❛r P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ♣r♦❣r❛♠ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ✐♥t P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ❜♦♦❧ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ tr✉❡ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ ❢❛❧s❡ P❛❧❛✈r❛ ❘❡s❡r✈❛❞❛ s❦✐♣
❉❡❧✐♠✐t❛❞♦r ✭ ❉❡❧✐♠✐t❛❞♦r ✮
❉❡❧✐♠✐t❛❞♦r ④ ❉❡❧✐♠✐t❛❞♦r ⑥
❋✐❣✉r❛ ✶✿ ❊s♣❡❝✐✜❝❛çã♦ ▲é①✐❝❛ ❞❡ ■▼P
❆❧é♠ ❞♦s t♦❦❡♥s ❛♣r❡s❡♥t❛❞♦s ♥❛ t❛❜❡❧❛ ❛♥t❡r✐♦r✱ ■▼P ♣♦ss✉✐ ♠❛✐s três t♦❦❡♥s ❞❡s❝r✐t♦s ❛❜❛✐①♦ ❝♦♠♦ ❡①♣r❡ssõ❡s r❡❣✉❧❛r❡s✳ ❖ t♦❦❡♥ ❧❡tr❛ r❡♣r❡s❡♥t❛ ✉♠❛ ❧❡tr❛ q✉❛❧q✉❡r✱ ❞✐❣✐t♦ ✉♠ ❞í❣✐t♦ ❡ ❛❧♣❤❛◆✉♠ q✉❛❧q✉❡r ❧❡tr❛ ♦✉ ❞í❣✐t♦✳
• ■❞❡♥t✐✜❝❛❞♦r❡s✿ ❧❡tr❛✭❧❡tr❛ ✰ ❞✐❣✐t♦✮∗ • ❈♦♥st❛♥t❡s ✐♥t❡✐r❛s✿ ❞✐❣✐t♦+
• ▲✐t❡r❛✐s ❞❡ ❙tr✐♥❣✿ ✧❛❧♣❤❛◆✉♠∗✧
✸✳✷ ❊s♣❡❝✐✜❝❛çã♦ ❙✐♥tát✐❝❛ ❞❛ ▲✐♥❣✉❛❣❡♠ ■▼P
◆❡st❛ s❡çã♦ é ❛♣r❡s❡♥t❛❞❛ ❛ ❣r❛♠át✐❝❛ ❞❛ ❧✐♥❣✉❛❣❡♠ ■▼P✳ ◆❡❧❛ ❛❞♦t❛♠♦s ❛s s❡❣✉✐♥t❡s ❝♦♥✈❡♥çõ❡s✿
• ❚❡r♠✐♥❛✐s✿ ❚♦❦❡♥s ✭t❛♠❜é♠ ❝❤❛♠❛❞♦s ❞❡ t❡r♠✐♥❛✐s✮ sã♦ r❡♣r❡s❡♥t❛✲ ❞♦s ✉t✐❧✐③❛♥❞♦ ❢♦♥t❡ t②♣❡✇r✐t❡r✳ ❖s t❡r♠✐♥❛✐s ✐❞❡♥t✐❢✐❡r✱ ♥✉♠❜❡r ❡ str✐♥❣ ❝♦rr❡s♣♦♥❞❡♠ ❛♦s t♦❦❡♥s ❛♣r❡s❡♥t❛❞♦s ✉t✐❧✐③❛♥❞♦ ❡①♣r❡ssõ❡s r❡❣✉❧❛r❡s ♥❛ s❡çã♦ ✸✳✶✳
• ◆ã♦ ❚❡r♠✐♥❛✐s✿ ❱❛r✐á✈❡✐s ❞❛ ❣r❛♠át✐❝❛ ✭t❛♠❜é♠ ❝❤❛♠❛❞♦s ❞❡ ♥ã♦ t❡r♠✐♥❛✐s✮ sã♦ r❡♣r❡s❡♥t❛❞♦s ✉t✐❧✐③❛♥❞♦ ❢♦♥t❡ ❡♠ ✐tá❧✐❝♦✳
❆ ❣r❛♠át✐❝❛ é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✷✳
❝♦♠♣✐❧❛t✐♦♥ ✉♥✐t → ✈❛r❜❧♦❝❦ ♣r♦❣r❛♠❜❧♦❝❦ ✈❛r❜❧♦❝❦ → ✈❛r ④ ✈❛r ❞❡❢s ⑥ ✈❛r ❞❡❢s → ✈❛r ❞❡❢ ✈❛r ❞❡❢s
| λ
✈❛r ❞❡❢ → t②♣❡ ✐❞❡♥t✐❢✐❡r ❀ t②♣❡ → ✐♥t |❜♦♦❧
♣r♦❣r❛♠❜❧♦❝❦ → ♣r♦❣r❛♠ ④ ❝♦♠♠❛♥❞s ⑥ ❝♦♠♠❛♥❞s → ❝♦♠♠❛♥❞ ❀ ❝♦♠♠❛♥❞s
| λ
❝♦♠♠❛♥❞ → s❦✐♣
| ✐❞❡♥t✐❢✐❡r ✿❂ ❜❡①♣ | ♣r✐♥t ❜❡①♣
| r❡❛❞ ✐❞❡♥t✐✜❡r
| ✐❢ ❜❡①♣ t❤❡♥ ❜❧♦❝❦ ❡❧s❡ ❜❧♦❝❦ | ✇❤✐❧❡ ❜❡①♣ ❞♦ ❜❧♦❝❦
❜❧♦❝❦ → ④ ❝♦♠♠❛♥❞s ⑥ ❜❡①♣ → tr✉❡ |❢❛❧s❡
| ❜❡①♣ ❂ ❜❡①♣ | ❜❡①♣ ❁❂ ❜❡①♣ | !❜❡①♣ | ❜❡①♣ ✫ ❜❡①♣ | ❜❡①♣ |❜❡①♣ | ✭ ❜❡①♣ ✮ | str✐♥❣ | ❛❡①♣ ❛❡①♣ → ♥✉♠❜❡r
| ✐❞❡♥t✐❢✐❡r | ❛❡①♣ ✰ ❛❡①♣ | ❛❡①♣ ✲ ❛❡①♣ | ❛❡①♣ ✯ ❛❡①♣ | ❛❡①♣ ✴ ❛❡①♣ | ✭ ❛❡①♣ ✮
❋✐❣✉r❛ ✷✿ ●r❛♠át✐❝❛ ▲✐✈r❡ ❞❡ ❈♦♥t❡①t♦ ❞❡ ■▼P
✸✳✸ ❱❡r✐✜❝❛çã♦ ❞❡ ❚✐♣♦s ❞❛ ▲✐♥❣✉❛❣❡♠ ■▼P
❆ ❧✐♥❣✉❛❣❡♠ ■▼P é ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❢♦rt❡♠❡♥t❡ t✐♣❛❞❛ ❡ ♥ã♦ ♣♦ss✉✐ ♣♦❧✐♠♦r✲ ✜s♠♦✳ ■▼P ♥ã♦ ♣♦ss✉✐ ✐♥❢❡rê♥❝✐❛ ❞❡ t✐♣♦s ❡✱ ♣♦rt❛♥t♦✱ t♦❞❛s ❛s s✉❛s ✈❛r✐á✈❡✐s ❞❡✈❡♠ s❡r ❞❡❝❧❛r❛❞❛s ❡ ❛♥♦t❛❞❛s ❝♦♠ ♦ s❡✉ r❡s♣❡❝t✐✈♦ t✐♣♦✳
P❛r❛ ❢♦r♠❛❧✐③❛r ❝♦♠♦ é ❢❡✐t❛ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ t✐♣♦s ❞❡ ✉♠❛ ❧✐♥❣✉❛❣❡♠ sã♦ ✉t✐❧✐③❛❞❛s r❡❣r❛s ❞❡ ❞❡❞✉çã♦✳ ❘❡❣r❛s ❞❡ ❞❡❞✉çã♦ sã♦ ❢♦r♠❛❧♠❡♥t❡ r❡♣r❡s❡♥✲ t❛❞❛s ♣♦r ✉♠❛ tr✐♣❧❛(Γ, e, τ)✱ ♦♥❞❡✿
• Γé ✉♠ ❝♦♥❥✉♥t♦ ❞❡♥♦♠✐♥❛❞♦ ❝♦♥t❡①t♦✱ ❢♦r♠❛❞♦ ♣♦r ♣❛r❡s(v, τ) ♦♥❞❡
vé ✉♠❛ ✈❛r✐á✈❡❧ ❡τ✱ ✉♠ t✐♣♦✳ ❯s✉❛❧♠❡♥t❡✱ ♦ ♣❛r(v, τ) é r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ v : τ✳ ❖ ❝♦♥t❡①t♦ ❛r♠❛③❡♥❛ ❛ ✐♥❢♦r♠❛çã♦ ❞❡ t✐♣♦s ❞❡ ✈❛r✐á✈❡✐s
❞❡❝❧❛r❛❞❛s ♥♦ ♣r♦❣r❛♠❛✳
• eé ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r ❞❛ s✐♥t❛①❡ ❞❛ ❧✐♥❣✉❛❣❡♠✳
• τ é ✉♠ t✐♣♦ q✉❛❧q✉❡r✳
❖ s✐❣♥✐✜❝❛❞♦ ❞❛ tr✐♣❧❛ (Γ, e, τ) é q✉❡ ♦ ❡❧❡♠❡♥t♦ e ♣♦ss✉✐ ♦ t✐♣♦ τ q✉❛♥❞♦
❛s ✈❛r✐á✈❡✐s ❞❡ e♣♦ss✉❡♠ ♦s t✐♣♦s ❡s♣❡❝✐✜❝❛❞♦s ❡♠ Γ✳ ❯s✉❛❧♠❡♥t❡ ❛ tr✐♣❧❛ (Γ, e, τ)é r❡♣r❡s❡♥t❛❞❛ ❝♦♠♦ Γ ⊢ e :τ✳
❉❡✈✐❞♦ ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❛ ❧✐♥❣✉❛❣❡♠ ■▼P✱ ♦ s❡✉ s✐st❡♠❛ ❞❡ t✐♣♦s é ❡①✲ tr❛♠❛♠❡♥t❡ ❞✐r❡t♦ ❡ ✐♥t✉✐t✐✈♦✳ P❛r❛ ❡s♣❡❝✐✜❝❛çã♦ ❞❛s r❡❣r❛s✱ ✉t✐❧✐③❛r❡♠♦s ✉♠❛ ♥♦t❛çã♦ ✐❞ê♥t✐❝❛ à ✉t✐❧✐③❛❞❛ ❡♠ ❧ó❣✐❝❛✳ ❈♦♠♦ ❡①❡♠♣❧♦ ❞❡st❛ ♥♦t❛çã♦✱ ❝♦♥s✐❞❡r❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ r❡❣r❛ ♠♦❞✉s✲♣♦♥❡♥s✿
a→b a ▼♦❞✉s P♦♥❡♥s
b
❊st❛ r❡❣r❛ ❞✐③ q✉❡ s❡a→bé ✈❡r❞❛❞❡ ❡aé ✈❡r❞❛❞❡✱ ♣♦❞❡♠♦s ❞❡❞✉③✐r q✉❡ b
é ✈❡r❞❛❞❡✐r♦✳
✸✳✸✳✶ ❱❡r✐✜❝❛çã♦ ❞❡ ❚✐♣♦s ❞❡ ❊①♣r❡ssõ❡s
❊①♣r❡ssõ❡s ❞❛ ❧✐♥❣✉❛❣❡♠ ■▼P sã♦ ❡❧❡♠❡♥t♦s s✐♥tát✐❝♦s ❞❛ ❧✐♥❣✉❛❣❡♠ q✉❡ ♣♦✲ ❞❡♠ s❡r ❞❡r✐✈❛❞♦s ✉t✐❧✐③❛♥❞♦ ❛s r❡❣r❛s ❜❡①♣ ❡ ❛❡①♣ ❞❛ ❣r❛♠át✐❝❛ ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✷✳
❖ r❡st❛♥t❡ ❞❡st❛ s❡çã♦ ✐rá ❛♣r❡s❡♥t❛r ❛ ❡s♣❡❝✐✜❝❛çã♦ ❞❛s r❡❣r❛s ♣❛r❛ ❡①♣r❡ssõ❡s ❞♦ s✐st❡♠❛ ❞❡ t✐♣♦s ❞❡ ■▼P✳ ❆ ♣r✐♠❡✐r❛ r❡❣r❛ ❞❡st❡ s✐st❡♠❛ ♥♦s ❞✐③ ❝♦♠♦ ✈❡r✐✜❝❛r ♦ t✐♣♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ ❊✈✐❞❡♥t❡♠❡♥t❡✱ t♦❞❛ ❝♦♥st❛♥t❡ ✐♥t❡✐r❛ ♣♦ss✉✐ ♦ t✐♣♦ ✐♥t✳ ❚❛❧ ❢❛t♦ é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ s❡❣✉✐♥t❡ r❡❣r❛✿
❚②✲◆✉♠❜❡r
Γ ⊢ n : int
❋✐❣✉r❛ ✸✿ ❘❡❣r❛s ♣❛r❛ ❚✐♣❛❣❡♠ ❞❡ ❈♦♥st❛♥t❡s ■♥t❡✐r❛s
❚②✲❚r✉❡
Γ ⊢ true : bool
❚②✲❋❛❧s❡
Γ ⊢ f alse : bool
❋✐❣✉r❛ ✹✿ ❘❡❣r❛s ♣❛r❛ ❚✐♣❛❣❡♠ ❞❡ ❈♦♥st❛♥t❡s ❇♦♦❧❡❛♥❛s
■st♦ é✱ s❡ n é ✉♠❛ ❝♦♥st❛♥t❡ ✐♥t❡✐r❛ q✉❛❧q✉❡r✱ ❡♥tã♦ s❡✉ t✐♣♦ ❞❡✈❡ s❡r
✐♥t✳ ❖ ♠❡s♠♦ ♣r✐♥❝í♣✐♦ é ✈á❧✐❞♦ ♣❛r❛ ❝♦♥st❛♥t❡s ❜♦♦❧❡❛♥❛s✱ ❝✉❥❛s r❡❣r❛s sã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛ ✜❣✉r❛ ✹✳
❆ ✈❡r✐✜❝❛çã♦ ❞❡ t✐♣♦s ♣❛r❛ ❡①♣r❡ssõ❡s ❛r✐t♠ét✐❝❛s é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✺✳ ❖❜s❡r✈❡ q✉❡ ♥❡st❛s r❡❣r❛s ♦ t✐♣♦ ❞❛ ❡①♣r❡ssã♦ é ❞❡✜♥✐❞♦ ❝♦♠ ❜❛s❡ ♥♦s t✐♣♦s ❞❛s s✉❜✲❡①♣r❡ssõ❡s q✉❡ ❛ ❝♦♠♣õ❡✳ ■st♦ é✱ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡ ♦ t✐♣♦ ❞❡
e1 +e2 é ✐♥t✱ t❡♠♦s q✉❡ ♣r✐♠❡✐r❛♠❡♥t❡ ❝♦♥❝❧✉✐r q✉❡ ♦ t✐♣♦ ❞❡ e1 ❡ ❞❡ e2 é
✐♥t✳ ❈❛s♦ ❝♦♥trár✐♦ t❡♠♦s ✉♠ ❡rr♦ ❞❡ t✐♣♦✳ Γ ⊢ e1 : int Γ ⊢ e2 : int
❚②✲P❧✉s
Γ ⊢ e1 + e2 :int
Γ ⊢ e1 : int Γ ⊢ e2 : int
❚②✲▼✐♥✉s
Γ ⊢ e1 − e2 :int
Γ ⊢ e1: int Γ ⊢ e2: int
❚②✲▼✉❧t
Γ ⊢ e1 ∗ e2 :int
Γ ⊢ e1: int Γ ⊢ e2 : int ❚②✲❉✐✈
Γ ⊢ e1/ e2 :int
❋✐❣✉r❛ ✺✿ ❘❡❣r❛s ♣❛r❛ ❚✐♣❛❣❡♠ ❞❡ ❊①♣r❡ssõ❡s ❆r✐t♠ét✐❝❛s
❆ ♠❡s♠❛ ❧✐♥❤❛ ❞❡ r❛❝✐♦❝í♥✐♦ ✈❛❧❡ ♣❛r❛ ❡①♣r❡ssõ❡s ❞❡ ❝♦♠♣❛r❛çã♦ ❡ ♦♣❡✲ r❛❞♦r❡s ❧ó❣✐❝♦s✱ ❝✉❥❛s r❡❣r❛s sã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛s ✜❣✉r❛s ✻ ❡ ✼✳
Γ ⊢ e1 : int Γ ⊢ e2 : int ❚②✲❊q
Γ ⊢ e1 = e2 :bool
Γ ⊢ e1: int Γ ⊢ e2 : int
❚②✲▲❡q
Γ ⊢ e1 ≤ e2 :bool
❋✐❣✉r❛ ✻✿ ❚✐♣❛❣❡♠ ❞❡ ❖♣❡r❛❞♦r❡s ❞❡ ❈♦♠♣❛r❛çã♦
❆ ú❧t✐♠❛ r❡❣r❛ ♣❛r❛ t✐♣❛❣❡♠ ❞❡ ❡①♣r❡ssõ❡s é ❛ ❞❡ ✈❛r✐á✈❡✐s✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ✈❛r✐á✈❡❧ v ♣♦ss✉✐ ♦ t✐♣♦ τ s❡ v : τ ∈ Γ✳ ❚❛❧ ❢❛t♦ é ❢♦r♠❛❧✐③❛❞♦ ♣❡❧❛ s❡❣✉✐♥t❡ r❡❣r❛ ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✽✿
Γ ⊢ e1: bool Γ ⊢ e2 : bool ❚②✲❖r
Γ ⊢ e1||e2 :bool
Γ ⊢ e1 : bool Γ ⊢ e2 : bool ❚②✲❆♥❞
Γ ⊢ e1&&e2 :bool
Γ ⊢ e: bool ❚②✲◆♦t
Γ ⊢!e:bool
❋✐❣✉r❛ ✼✿ ❘❡❣r❛s ♣❛r❛ ❚✐♣❛❣❡♠ ❞❡ ❖♣❡r❛❞♦r❡s ▲ó❣✐❝♦s Γ(v) =τ
❚②✲❱❛r
Γ ⊢ v:τ
❋✐❣✉r❛ ✽✿ ❘❡❣r❛ ♣❛r❛ ❚✐♣❛❣❡♠ ❞❡ ❱❛r✐á✈❡✐s
❆ ♥♦t❛çã♦ Γ(v) r❡♣r❡s❡♥t❛ ♦ t✐♣♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ✈❛r✐á✈❡❧ v✳ ❋♦r♠❛❧✲
♠❡♥t❡✱ t❡♠♦s q✉❡✿ Γ(v) ={τ|v:τ ∈Γ}✳ ✸✳✸✳✷ ❱❡r✐✜❝❛çã♦ ❞❡ ❚✐♣♦s ❞❡ ❈♦♠❛♥❞♦s
❯♠❛ ✈❡③ q✉❡ ❝♦♠❛♥❞♦s ❡①❡❝✉t❛♠ ✉♠❛ ❛çã♦ q✉❡ ♣♦❞❡ r❡s✉❧t❛r ❡♠ ✉♠ ❡❢❡✐t♦ ❝♦❧❛t❡r❛❧ ✭✐♠♣r❡ssã♦✱ ❛❧t❡r❛çã♦ ❞♦ ❡st❛❞♦ ♦✉ ❧❡✐t✉r❛ ❞❡ ✈❛❧♦r❡s✮ ❡st❡s ♥ã♦ ♣♦ss✉❡♠ t✐♣♦s✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❝♦♠❛♥❞♦cé ❜❡♠ ❢♦r♠❛❞♦ ❡♠ ✉♠ ❝♦♥t❡①t♦
Γ✱Γ|= c✱ s❡ t♦❞❛ ❡①♣r❡ssã♦eq✉❡ ❝♦♠♣õ❡cé ❜❡♠ t✐♣❛❞❛✱ ✐st♦ éΓ ⊢ e: τ✳
❆s r❡❣r❛s ♣❛r❛ ✈❡r✐✜❝❛çã♦ s❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝♦♠❛♥❞♦ é ❜❡♠ ❢♦r♠❛❞♦ sã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛ ✜❣✉r❛ ✾✳
❚②✲❙❦✐♣
Γ|= s❦✐♣
Γ⊢ v: τ ❚②✲❘❡❛❞
Γ|= r❡❛❞ ✈
Γ⊢ e:τ ❚②✲Pr✐♥t
Γ|= ♣r✐♥t ❡
Γ⊢ v:τ Γ⊢ e:τ ❚②✲❆ss✐❣♥
Γ|= ✈✿❂❡
❋✐❣✉r❛ ✾✿ ❘❡❣r❛s ♣❛r❛ ❱❡r✐✜❝❛çã♦ ❞❡ ❈♦♠❛♥❞♦s ❇❡♠ ❋♦r♠❛❞♦s ✖ P❛rt❡ ✶ P❛r❛ ✈❡r✐✜❝❛çã♦ ❞❡ ❝♦♠❛♥❞♦s ❝♦♥❞✐❝✐♦♥❛✐s✱ ❧♦♦♣s ♦✉ ♠❡s♠♦ ♣r♦❣r❛♠❛s ❜❡♠ ❢♦r♠❛❞♦s ❞❡✈❡♠♦s ♥♦s ✈❛❧❡r ❞❡ ✉♠❛ r❡❣r❛ ♣❛r❛ ✈❡r✐✜❝❛çã♦ ❞❡ ✉♠ ❝♦♥✲
❥✉♥t♦s ❞❡ ❝♦♠❛♥❞♦s ❜❡♠ ❢♦r♠❛❞♦s✳ ❯♠ ❜❧♦❝♦ ❞❡n≥0 ❝♦♠❛♥❞♦s ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡n ❝♦♠❛♥❞♦s✳ ❉❡♥♦t❛♠♦s q✉❡ ✉♠ ❝♦♥❥✉♥t♦C ❞❡
❝♦♠❛♥❞♦s é ❜❡♠ ❢♦r♠❛❞♦ ♣♦rΓ|=set
C✳ ❊✈✐❞❡♥t❡♠❡♥t❡✱ ✉♠ ❝♦♥❥✉♥t♦ ✈❛③✐♦
❞❡ ❝♦♠❛♥❞♦s é ❜❡♠ ❢♦r♠❛❞♦✳ ❚❛❧ ❢❛t♦ é ❡①♣r❡ss♦ ♣❡❧❛ s❡❣✉✐♥t❡ r❡❣r❛✿
❚②✲❊♠♣t②✲❇❧♦❝❦
Γ|=set
∅
❋✐❣✉r❛ ✶✵✿ ❘❡❣r❛ ♣❛r❛ ❱❡r✐✜❝❛çã♦ ❞❡ ✉♠ ❈♦♥❥✉♥t♦ ❱❛③✐♦ ❞❡ ❈♦♠❛♥❞♦s P❛r❛ ✈❡r✐✜❝❛r q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♠❛♥❞♦s C 6= ∅ é ❜❡♠ ❢♦r♠❛❞♦✱ ❜❛st❛ ❧❡♠❜r❛r q✉❡ ❞❡✈❡ ❡①✐st✐r ✉♠ ❝♦♥❥✉♥t♦ C′ ❡ ✉♠ ❝♦♠❛♥❞♦ c t❛❧ q✉❡
C = {c} ∪ C′✳ P❛r❛ C s❡r ❜❡♠ ❢♦r♠❛❞♦✱ t❡♠♦s q✉❡ ♦ ❝♦♠❛♥❞♦ c ❞❡✈❡ s❡r ❜❡♠ ❢♦r♠❛❞♦ ❡ ♦ ❝♦♥❥✉♥t♦C′ t❛♠❜é♠✳ ■st♦ é ❡①♣r❡ss♦ ♣❡❧❛ r❡❣r❛✿
Γ|= c Γ|=set C′
❚②✲◆♦♥✲❊♠♣t②✲❇❧♦❝❦
Γ|=set
{c} ∪ C′
❋✐❣✉r❛ ✶✶✿ ❘❡❣r❛ ♣❛r❛ ❱❡r✐✜❝❛çã♦ ❞❡ ✉♠ ❈♦♥❥✉♥t♦ ♥ã♦ ❱❛③✐♦ ❞❡ ❈♦♠❛♥❞♦s ❉❡ ♣♦ss❡ ❞❛s r❡❣r❛s ♣❛r❛ ✈❡r✐✜❝❛çã♦ ❞❡ ❝♦♠❛♥❞♦s✱ ♣♦❞❡♠♦s ❢♦r♠❛❧✐③❛r q✉❛♥❞♦ ♦ ❝♦♠❛♥❞♦s ❝♦♥❞✐❝✐♦♥❛✐s ❡ ❧♦♦♣s sã♦ ❜❡♠ ❢♦r♠❛❞♦s✳
Γ⊢ e:bool Γ|=set
C1 Γ|=set C2
❚②✲■❢
Γ|= ✐❢ ❡ t❤❡♥ C1 ❡❧s❡ C2
❋✐❣✉r❛ ✶✷✿ ❘❡❣r❛ ♣❛r❛ ❱❡r✐✜❝❛çã♦ ❞❡ ❈♦♠❛♥❞♦s ✖ P❛rt❡ ✷
❆ r❡❣r❛ ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✶✷ é ❞❡ s✐♠♣❧❡s ❝♦♠♣r❡❡sã♦✳ ❯♠ ❝♦♠❛♥❞♦ ✐❢ é ❜❡♠ ❢♦r♠❛❞♦ s❡ ❛ ❡①♣r❡ssã♦ ❞♦ t❡st❡ ♣♦ss✉✐ ♦ t✐♣♦ ❜♦♦❧ ❡ ♦s ❜❧♦❝♦s ❞❡ ❝♦♠❛♥❞♦ ♣❛r❛ ♦ t❤❡♥ ❡ ♦ ❡❧s❡ sã♦ ❜❡♠ ❢♦r♠❛❞♦s✳ ❆ r❡❣r❛ ♣❛r❛ ✈❡r✐✜❝❛çã♦ ❞❡ ❝♦♠❛♥❞♦s ✇❤✐❧❡ é s✐♠✐❧❛r ❡ é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✶✸✳
❈♦♥❢♦r♠❡ ❥á ❛♣r❡s❡♥t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♣r♦❣r❛♠❛s ■▼P ❝♦♥s✐st❡♠ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♠❛♥❞♦s ❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❡✜♥✐çõ❡s ❞❡ ✈❛r✐á✈❡✐s✳ ❯♠ ♣r♦❣r❛♠❛ ■▼P é ❝♦rr❡t♦ s❡ ❡ s♦♠❡♥t❡ s❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♠❛♥❞♦s é ❜❡♠ ❢♦r✲ ♠❛❞♦ ✉t✐❧✐③❛♥❞♦ ❝♦♠♦ ❝♦♥t❡①t♦ ❛s ❞❡✜♥✐çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❞❡❝❧❛r❛❞❛s✳ P♦❞❡✲ ♠♦s ❝♦♥s✐❞❡r❛r q✉❡ t♦❞♦ ♣r♦❣r❛♠❛ ■▼Ppé ❢♦r♠❛❞♦ ♣♦r ✉♠ ♣❛rp= (Γ, C)✱ ♦♥❞❡Γ ={vi:τi}i=1..né ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❡✜♥✐çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❡
C ♦ ❝♦♥❥✉♥t♦
❞❡ ❝♦♠❛♥❞♦s q✉❡ ❞❡✜♥❡ ♦ ♣r♦❣r❛♠❛✳ P♦❞❡♠♦s ❛✜r♠❛r q✉❡ ✉♠ ♣r♦❣r❛♠❛ ■▼P ❡stá ❝♦rr❡t♦ s❡ ❡ s♦♠❡♥t❡ s❡Γ|=set
C✱ ♦♥❞❡ p= (Γ, C)✳
Γ⊢ e:bool Γ|=set C
❚②✲❲❤✐❧❡
Γ|= ✇❤✐❧❡ ❡ ❞♦C
❋✐❣✉r❛ ✶✸✿ ❘❡❣r❛ ♣❛r❛ ❱❡r✐✜❝❛çã♦ ❞❡ ❈♦♠❛♥❞♦s ✖ P❛rt❡ ✸
✸✳✹ ❙❡♠â♥t✐❝❛ ❖♣❡r❛❝✐♦♥❛❧ ❞❡ Pr♦❣r❛♠❛s ■▼P
P❛r❛ ❢♦r♠❛❧✐③❛r ❛ ❡①❡❝✉çã♦ ❞❡ ♣r♦❣r❛♠❛s✱ ❞❡✈❡♠♦s s❡r ❝❛♣❛③❡s ❞❡ ❛tr✐❜✉✐r ❛ ❡st❡s ✉♠ s✐❣♥✐✜❝❛❞♦ ♣r❡❝✐s♦✳ ❯♠❛ ❞❛s ♠❛♥❡✐r❛s ❞❡ s❡ ❢♦r♠❛❧✐③❛r ♦ s✐❣♥✐✜✲ ❝❛❞♦ ❞❡ ♣r♦❣r❛♠❛s é ✐♥t❡r♣r❡tá✲❧♦s ❡♠ ✉♠❛ ♠áq✉✐♥❛ ❛❜str❛t❛✳ ❆ s❡♠â♥t✐❝❛ ♦♣❡r❛❝✐♦♥❛❧ é ✉♠ ❢♦r♠❛❧✐s♠♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❡s♣❡❝✐✜❝❛r ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ♣r♦❣r❛♠❛s ❡①✐❜✐♥❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡st❡s ✉t✐❧✐③❛♥❞♦ tr❛♥s✐çõ❡s ❡♥tr❡ ❞✐✲ ❢❡r❡♥t❡s ❡st❛❞♦s ❞❡ ✉♠❛ ♠áq✉✐♥❛ ❛❜str❛t❛✳
Pr✐♠❡✐r❛♠❡♥t❡✱ ❞❡✈❡♠♦s ❞❡✜♥✐r ♦ q✉❡ s❡rã♦ ♦s ❡st❛❞♦s ❡ ❛s tr❛♥s✐çõ❡s ❞❛ ♠áq✉✐♥❛ q✉❡ ❡①❡❝✉t❛rá ♣r♦❣r❛♠❛s ■▼P✳ ❊♠ t♦❞❛ ❛ ❧✐♥❣✉❛❣❡♠ ✐♠♣❡r❛t✐✈❛✱ ❛ ♥♦çã♦ ❞❡ ❡st❛❞♦ é ❝❛r❛❝t❡r✐③❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ❡ s❡✉s ✈❛❧♦r❡s ❛ss♦❝✐❛❞♦s ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ tr❡❝❤♦ ❞❡ ❝ó❞✐❣♦✳ ❉❡st❛ ♠❛♥❡✐r❛✱ ♣♦❞❡♠♦s ♣❡♥s❛r q✉❡ ♦ ❡st❛❞♦ é ❞❛❞♦ ♣♦r ✉♠❛ ❢✉♥çã♦ σ : V → N⊕ {true, f alse} ♦♥❞❡✿
• V ✿ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s✳
• N✿ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s
• ⊕✿ ❯♥✐ã♦ ❞✐s❥✉♥t❛✳ ❙❡❥❛♠ A ❡ B ❞♦✐s ❝♦♥❥✉♥t♦s q✉❛✐sq✉❡r✱ ❞❡✜♥❡✲s❡ A⊕B ❝♦♠♦✿ A⊕B={0} ×A∪ {1} ×B✳
■♥❢♦r♠❛❧♠❡♥t❡✱ σ(v) r❡t♦r♥❛ ♦ ✈❛❧♦r ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ✈❛r✐á✈❡❧ v ❞❡ ❛❝♦r❞♦
❝♦♠ ♦ t✐♣♦ ❞❡st❛✳ ❙❡ Γ(v) = int✱ ❡♥tã♦ σ(v) ∈N✳ ❈❛s♦ Γ(v) = bool✱ ❡♥tã♦
σ(v)∈ {true, f alse}✳
❉❡ ♣♦ss❡ ❞❛ ❞❡✜♥✐çã♦ ❞♦ ❡st❛❞♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ q✉❡ sã♦ tr❛♥s✐çõ❡s✳ ❉❡♥♦♠✐♥❛r❡♠♦s ♣♦r ❝♦♥✜❣✉r❛çã♦ ♦ ♣❛r ha, σi ❢♦r♠❛❞♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ a
q✉❛❧q✉❡r ❞❛ s✐♥t❛①❡ ❞❡ ■▼P ❡ ❛ ❢✉♥çã♦ q✉❡ ❞❡✜♥❡ ♦ ❡st❛❞♦✱σ✳
❆ s❡♠â♥t✐❝❛ ♦♣❡r❛❝✐♦♥❛❧ ❞❡ ■▼P ❝♦♥s✐st❡ ❞❡ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❝♦♥✜❣✉✲ r❛çõ❡s ❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡ s❡ ❡①❡❝✉t❛r ♦ ❡❧❡♠❡♥t♦ s✐♥tát✐❝♦ ❞❡ss❛ ❝♦♥✜❣✉r❛çã♦✳ ❙❡ ❝♦♥s✐❞❡r❛r♠♦s ❡①♣r❡ssõ❡s ❛r✐t♠ét✐❝❛s✱ t❡♠♦s q✉❡ ♦ r❡s✉❧t❛❞♦ ❞❛ ❡①❡❝✉çã♦ é ✉♠ ♥ú♠❡r♦❀ ♣❛r❛ ❡①♣r❡ssõ❡s ❜♦♦❧❡❛♥❛s✱ t❡♠♦s q✉❡ ♦ r❡s✉❧t❛❞♦ é ✉♠ ✈❛❧♦r ❧ó❣✐❝♦✳ ❏á ❛ ❡①❡❝✉çã♦ ❞❡ ❝♦♠❛♥❞♦s✱ ♣r♦❞✉③ ❝♦♠♦ r❡s✉❧t❛❞♦ ✉♠❛ ♥♦✈♦ ❡st❛❞♦✳ ✸✳✹✳✶ ❙❡♠â♥t✐❝❛ ❖♣❡r❛❝✐♦♥❛❧ ❞❡ ❊①♣r❡ssõ❡s ❆r✐t♠ét✐❝❛s
Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❡①❡❝✉çã♦ ❞❡ ✉♠❛ ❡①♣r❡ssã♦ ❛r✐t♠ét✐❝❛
e❡♠ ✉♠ ❡st❛❞♦ σ✳ P❛r❛ ❡①♣r❡ssõ❡s ❛r✐t♠ét✐❝❛s✱ ❛ s❡♠â♥t✐❝❛ ♦♣❡r❛❝✐♦♥❛❧ é
❞❡✜♥✐❞❛ ❝♦♠♦ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❝♦♥✜❣✉r❛çõ❡s ❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❘❡♣r❡✲ s❡♥t❛r❡♠♦s t❛❧ r❡❧❛çã♦✱ ✉t✐❧✐③❛♥❞♦ ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦✿
ha, σi →n
❊st❛ r❡❧❛çã♦ s❡rá ❞❡✜♥✐❞❛ ✐♥❞✉t✐✈❛♠❡♥t❡ s♦❜r❡ ❛ s✐♥t❛①❡ ❞❡ ❡①♣r❡ssõ❡s ❛r✐t✲ ♠ét✐❝❛s✳ ❈♦♠♦ ♣r✐♠❡✐r♦ ❝❛s♦ ❝♦♥s✐❞❡r❛r❡♠♦s ✉♠❛ ❡①♣r❡ssã♦ ❢♦r♠❛❞❛ ❛♣❡♥❛s ♣♦r ✉♠❛ ❝♦♥st❛♥t❡ ♥✉♠ér✐❝❛✳ ❆ ✜❣✉r❛ ✶✹ ❛♣r❡s❡♥t❛ ❛ r❡❣r❛ ♣❛r❛ s❡ ❛✈❛❧✐❛r ✉♠❛ ❝♦♥st❛♥t❡ ♥✉♠ér✐❝❛ ❡♠ ✉♠ ❡st❛❞♦ σ✳ ➱ ó❜✈✐♦ q✉❡ ❡①❡❝✉t❛r ✉♠❛ ❝♦♥s✲
t❛♥t❡ ♥ú♠❡r✐❝❛ ❡♠ ✉♠ ❡st❛❞♦ q✉❛❧q✉❡r ♣r♦❞✉③ ❝♦♠♦ r❡s✉❧t❛❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ❡st❛ ❡①♣r❡ssã♦✳
❊✈❛❧✲◆✉♠
hn, σi → n
❋✐❣✉r❛ ✶✹✿ ❆✈❛❧✐❛♥❞♦ ❈♦♥st❛♥t❡s ◆✉♠ér✐❝❛s
❈❛s♦ ❛ ❡①♣r❡ssã♦ ❛r✐t♠ét✐❝❛ s❡❥❛ ❞❛ ❢♦r♠❛e1◦e2✱ ♦♥❞❡e1, e2 sã♦ ❡①♣r❡s✲
sõ❡s ❛r✐t♠ét✐❝❛s ❡◦ ∈ {+,−,∗, /}✱ t❡♠♦s q✉❡ ❛ ❡①❡❝✉çã♦ ♣r♦❝❡❞❡ ❞❡ ❛❝♦r❞♦
❝♦♠ ❛ s❡❣✉✐♥t❡ r❡❣r❛✿
he1, σi → n1 he2, σi → n2
❊✈❛❧✲❇✐♥❆r✐t❤❖♣
he1◦e2, σi → n1◦n2
❋✐❣✉r❛ ✶✺✿ ❆✈❛❧✐❛♥❞♦ ❖♣❡r❛❞♦r❡s ❆r✐t♠ét✐❝♦s
✸✳✹✳✷ ❙❡♠â♥t✐❝❛ ❖♣❡r❛❝✐♦♥❛❧ ❞❡ ❊①♣r❡ssõ❡s ❇♦♦❧❡❛♥❛s
◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s ❝♦♠♦ ❡①❡❝✉t❛r ❡①♣r❡ssõ❡s ❜♦♦❧❡❛♥❛s✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❡st❛❞♦σ✳ ■♥❞❡♣❡❞❡♥t❡♠❡♥t❡ ❞♦ ❡st❛❞♦✱ ❛s ❝♦♥st❛♥t❡s
❧ó❣✐❝❛s tr✉❡ ❡ ❢❛❧s❡ sã♦ ❛✈❛❧✐❛❞❛s ♣❛r❛ ❡❧❛s ♣ró♣r✐❛s ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ❡st❛❞♦ ❝♦♥s✐❞❡r❛❞♦✳ ❆ r❡❣r❛ ♣❛r❛ ❛✈❛❧✐❛çã♦ ❞❡ ❝♦♥st❛♥t❡s ❜♦♦❧❡❛♥❛s é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✶✻✳
❊✈❛❧✲❚r✉❡
htr✉❡, σi →tr✉❡
❊✈❛❧✲❋❛❧s❡
h❢❛❧s❡, σi →❢❛❧s❡
❋✐❣✉r❛ ✶✻✿ ❆✈❛❧✐❛♥❞♦ ❈♦♥st❛♥t❡s ❇♦♦❧❡❛♥❛s ❆ s❡♠â♥t✐❝❛ ❞♦ ♦♣❡r❛❞♦r ❞❡ ✐❣✉❛❧❞❛❞❡ é ❞❛❞❛ ♣♦r ❞✉❛s r❡❣r❛s✿
✶✳ ❊✈❛❧✲❊q✲❚r✉❡✿ ❊st❛ r❡❣r❛ é ✉t✐❧✐③❛❞❛ ♣❛r❛ ❝♦♠♣✉t❛r ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❞✉❛s ❡①♣r❡ssõ❡s ✐❣✉❛✐s ❡✱ ♣♦rt❛♥t♦✱ ❛✈❛❧✐❛ ❛ ✐❣✉❛❧❞❛❞❡ ♣❛r❛ tr✉❡✳ ✷✳ ❊✈❛❧✲❊q✲❋❛❧s❡✿ ❊st❛ r❡❣r❛ é ✉t✐❧✐③❛❞❛ ♣❛r❛ ❝♦♠♣✉t❛r ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡
❆♠❜❛s ❛s r❡❣r❛s sã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛ ✜❣✉r❛ ✶✼✳ he1, σi → n he2, σi → n
❊✈❛❧✲❊q✲❚r✉❡
he1=e2, σi → tr✉❡
he1, σi → n he2, σi → m
❊✈❛❧✲❊q✲❋❛❧s❡
he1 =e2, σi → ❢❛❧s❡
❋✐❣✉r❛ ✶✼✿ ❆✈❛❧✐❛♥❞♦ ❛ ■❣✉❛❧❞❛❞❡
❆ss✐♠ ❝♦♠♦ ♦ ♦♣❡r❛❞♦r ❞❡ ✐❣✉❛❧❞❛❞❡✱ ♦ ♦♣❡r❛❞♦r ❞❡ ❝♦♠♣❛r❛çã♦ ❁❂ t❛♠✲ ❜é♠ ♣♦ss✉✐ ❞✉❛s r❡❣r❛s ♣❛r❛ ❛✈❛❧✐❛çã♦✳
he1, σi → n he2, σi → m ∃n0.n0+n=m
❊✈❛❧✲❈♠♣✲❚r✉❡
he1 ≤e2, σi → tr✉❡
he1, σi → n he2, σi → m 6 ∃n0.n0+n=m
❊✈❛❧✲❈♠♣✲❋❛❧s❡
he1 ≤e2, σi → ❢❛❧s❡
❋✐❣✉r❛ ✶✽✿ ❆✈❛❧✐❛♥❞♦ ♦ ❖♣❡r❛❞♦r ❞❡ ❈♦♠♣❛r❛çã♦ ❁❂
❆ ✜❣✉r❛ ✶✾ ❛♣r❡s❡♥t❛ ❛s r❡❣r❛s ♣❛r❛ ❛✈❛❧✐❛çã♦ ❞❛ ♥❡❣❛çã♦✱ ❛ ✜❣✉r❛ ✷✵ r❡❣r❛s ♣❛r❛ ❛✈❛❧✐❛çã♦ ❞❛ ❝♦♥❥✉♥çã♦ ❡ ❛ ✜❣✉r❛ ✷✶ ♣❛r❛ ❛ ❞✐s❥✉♥çã♦✳ ❚♦❞❛s ❡st❛s r❡❣r❛s ♣♦ss✉❡♠ s✐❣♥✐✜❝❛❞♦ ✐♥t✉✐t✐✈♦ ❡ ✐♠❡❞✐❛t♦✳
he, σi →tr✉❡
❊✈❛❧✲◆♦t✲❚r✉❡
h¬e, σi →❢❛❧s❡ he, σi →❢❛❧s❡
❊✈❛❧✲◆♦t✲❋❛❧s❡
h¬e, σi →tr✉❡
❋✐❣✉r❛ ✶✾✿ ❆✈❛❧✐❛♥❞♦ ♦ ❖♣❡r❛❞♦r ❞❡ ◆❡❣❛çã♦
✸✳✹✳✸ ❙❡♠â♥t✐❝❛ ❖♣❡r❛❝✐♦♥❛❧ ❞❡ ❊①♣r❡ssõ❡s ❊♥✈♦❧✈❡♥❞♦ ❱❛r✐á✈❡✐s ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❣r❛♠át✐❝❛ ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✷✱ ❡①♣r❡ssõ❡s ■▼P ♣♦❞❡♠ ♣♦ss✉✐r ✈❛r✐á✈❡✐s✳ ❆ ❛✈❛❧✐❛çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ é ❢❡✐t❛ ❝♦♥s✉❧t❛♥❞♦ ♦ s❡✉ ✈❛❧♦r ❛t✉❛❧ q✉❡ é ❞❛❞♦ ♣❡❧♦ ❡st❛❞♦ σ ❞♦ ♣r♦❣r❛♠❛✳ ❉❡st❛ ♠❛♥❡✐r❛✱ s❡ v é ✉♠❛
✈❛r✐á✈❡❧ ♦ r❡s✉❧t❛❞♦ ❞❡ ❛✈❛❧✐á✲❧❛ é σ(v)✱ ♦✉ s❡❥❛✱ ♦ r❡s✉❧t❛❞♦ ❞❡ ❛✈❛❧✐❛r ✉♠❛ ✈❛r✐á✈❡❧ é ♦ ❝♦♥t❡ú❞♦ ❞❡st❛ q✉❡ é ❛r♠❛③❡♥❛❞♦ ♣❡❧❛ ❢✉♥çã♦σ✳
❆ r❡❣r❛ ♣❛r❛ ❛✈❛❧✐❛r ✉♠❛ ✈❛r✐á✈❡❧ q✉❛❧q✉❡r é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✷✷✳
he1, σi →❢❛❧s❡ he2, σi →❢❛❧s❡
❊✈❛❧✲❆♥❞✲❋❛❧s❡✲❋❛❧s❡
he1∧e2, σi →❢❛❧s❡
he1, σi →tr✉❡ he2, σi →❢❛❧s❡
❊✈❛❧✲❆♥❞✲❚r✉❡✲❋❛❧s❡
he1∧e2, σi →❢❛❧s❡
he1, σi →❢❛❧s❡ he2, σi →tr✉❡
❊✈❛❧✲❆♥❞✲❋❛❧s❡✲❚r✉❡
he1∧e2, σi →❢❛❧s❡
he1, σi →tr✉❡ he2, σi →tr✉❡
❊✈❛❧✲❆♥❞✲❚r✉❡✲❚r✉❡
he1∧e2, σi →tr✉❡
❋✐❣✉r❛ ✷✵✿ ❆✈❛❧✐❛♥❞♦ ❛ ❈♦♥❥✉♥çã♦ he1, σi →❢❛❧s❡ he2, σi →❢❛❧s❡
❊✈❛❧✲❖r✲❋❛❧s❡✲❋❛❧s❡
he1∨e2, σi →❢❛❧s❡
he1, σi →❢❛❧s❡ he2, σi →tr✉❡
❊✈❛❧✲❖r✲❋❛❧s❡✲❚r✉❡
he1∨e2, σi →tr✉❡
he1, σi →tr✉❡ he2, σi →❢❛❧s❡
❊✈❛❧✲❖r✲❚r✉❡✲❋❛❧s❡
he1∨e2, σi →tr✉❡
he1, σi →tr✉❡ he2, σi →tr✉❡
❊✈❛❧✲❖r✲❚r✉❡✲❚r✉❡
he1∨e2, σi →tr✉❡
❋✐❣✉r❛ ✷✶✿ ❆✈❛❧✐❛♥❞♦ ❛ ❉✐s❥✉♥çã♦ ✸✳✹✳✹ ❙❡♠â♥t✐❝❛ ❖♣❡r❛❝✐♦♥❛❧ ❞❡ ❈♦♠❛♥❞♦s
❊①♣r❡ssõ❡s sã♦ ❛✈❛❧✐❛❞❛s ♣❛r❛ ✈❛❧♦r❡s ✭✐♥t❡✐r♦s ♦✉ ❧ó❣✐❝♦s✮ ❡♠ ✉♠ ❞❡t❡r♠✐✲ ♥❛❞♦ ❡st❛❞♦σ✱ ♥ã♦ ♣♦❞❡♥❞♦ ❛❧t❡r❛r s❡✉ ❝♦♥t❡ú❞♦✳ ❈♦♠❛♥❞♦s✱ ♣♦r s✉❛ ✈❡③✱ ❛♦
s❡r❡♠ ❡①❡❝✉t❛❞♦s ♣♦❞❡♠ ❛❧t❡r❛r ♦ ❡st❛❞♦✳ ❆♦ ✐♥✐❝✐❛r ❛ ❡①❡❝✉çã♦ ❞❡ ✉♠ ♣r♦✲ ❣r❛♠❛ ■▼P ❞❡✈❡♠♦s ❛ss✉♠✐r q✉❡ t♦❞❛s ❛s ✈❛r✐á✈❡✐s ✐♥t❡✐r❛s sã♦ ✐♥✐❝✐❛❧✐③❛❞❛s ❝♦♠0 ❡ ❧ó❣✐❝❛s ❝♦♠ ❢❛❧s❡✳
❯♠❛ ❝♦♥✜❣✉r❛çã♦ ♣❛r❛ ❛ ❡①❡❝✉çã♦ ❞❡ ❝♦♠❛♥❞♦s é ✉♠ ♣❛rhc, σi❢♦r♠❛❞♦ ♣♦r ✉♠ ❝♦♠❛♥❞♦ c ❡ ✉♠ ❡st❛❞♦ σ✳ P♦r s✉❛ ✈❡③✱ ❛ s❡♠â♥t✐❝❛ ❞❡ ❝♦♠❛♥❞♦s
❞❡✜♥❡ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❝♦♥✜❣✉r❛çõ❡s ❡ ❡st❛❞♦s✿ hc, σi →σ′
q✉❡ s✐❣♥✐✜❝❛ q✉❡ ❛ ❡①❡❝✉çã♦ ❝♦♠♣❧❡t❛ ❞♦ ❝♦♠❛♥❞♦c❡♠ ✉♠ ❡st❛❞♦σ t❡r♠✐♥❛
❊✈❛❧✲❱❛r
hv, σi → σ(v)
❋✐❣✉r❛ ✷✷✿ ❆✈❛❧✐❛♥❞♦ ✉♠❛ ❱❛r✐á✈❡❧
❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦✿ σ ◦[v 7→ m]✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ s✉❜st✐t✉✐çã♦ ❞♦ ✈❛❧♦r ❝♦rr❡♥t❡ ❞❛ ✈❛r✐á✈❡❧v♣♦rm✳ P♦r ❡①❡♠♣❧♦ ❝♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❝♦♥✜❣✉r❛çã♦
hv✿❂5, σi✱ ♦ r❡s✉❧t❛❞♦ ❞❡ ❛✈❛❧✐❛r ❡st❡ ❝♦♠❛♥❞♦ é ♦ ❡st❛❞♦ ♥♦ q✉❛❧ ❛ ✈❛r✐á✈❡❧
v ♣♦ss✉✐ ❝♦♠♦ ✈❛❧♦r ♦ ♥ú♠❡r♦5✱ ✐st♦ éσ◦[v7→5]✳
❆ ✜❣✉r❛ ✷✸ ♠♦str❛ ❛ r❡❣r❛ ♣❛r❛ ❛✈❛❧✐❛çã♦ ❞♦ ❝♦♠❛♥❞♦ s❦✐♣✱ q✉❡ ♥ã♦ r❡❛❧✐③❛ ❛❧t❡r❛çã♦ ♥♦ ❡st❛❞♦ ❞♦ s♦❢t✇❛r❡✳
❊✈❛❧✲❙❦✐♣
hs❦✐♣, σi →σ
❋✐❣✉r❛ ✷✸✿ ❆✈❛❧✐❛♥❞♦ ♦ ❈♦♠❛♥❞♦ ❙❦✐♣
❆ ❛✈❛❧✐❛çã♦ ❞❡ ✉♠❛ ❛tr✐❜✉✐çã♦ é ❞✐r❡t❛✳ ❇❛st❛ ♣r✐♠❡✐r❛♠❡♥t❡ ❛✈❛❧✐❛r ❛ ❡①♣r❡ssã♦ ❡ ♥❛ s❡q✉ê♥❝✐❛✱ r❡t♦r♥❛r ♦ ❡st❛❞♦ ♠♦❞✐✜❝❛❞♦✳
he, σi → m
❊✈❛❧✲❆tr✐❜
hv✿❂e, σi →σ◦[v7→m]
❋✐❣✉r❛ ✷✹✿ ❆✈❛❧✐❛♥❞♦ ❛ ❛tr✐❜✉✐çã♦
◗✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ❛ s❡q✉ê♥❝✐❛ ❞❡ ❞♦✐s ♦✉ ♠❛✐s ❝♦♠❛♥❞♦s✱ ♦ r❡s✉❧t❛❞♦ é ♦ ❡st❛❞♦ ♣r♦❞✉③✐❞♦ ♣❡❧♦ ú❧t✐♠♦ ❝♦♠❛♥❞♦ ❡①❡❝✉t❛❞♦✳ ❊✈✐❞❡♥t❡♠❡♥t❡✱ ❛♣ós ❛ ❡①❡❝✉çã♦ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝♦♠❛♥❞♦✱ ♦ ❡st❛❞♦ ♣r♦❞✉③✐❞♦ é ❡♥✈✐❛❞♦ ♣❛r❛ ♦ ♣ró①✐♠♦ ❝♦♠❛♥❞♦ ♥❛ s❡q✉ê♥❝✐❛✳ ❚❛❧ ❢❛t♦ é ❢♦r♠❛❧✐③❛❞♦ ♣❡❧❛ s❡❣✉✐♥t❡ r❡❣r❛ ❞❛ ✜❣✉r❛ ✷✺✳
❆ ❛✈❛❧✐❛çã♦ ❞❡ ❝♦♠❛♥❞♦s ❝♦♥❞✐❝✐♦♥❛✐s ❡ ❧♦♦♣s ❞❡♣❡♥❞❡ ❞♦ r❡s✉❧t❛❞♦ ❞❛ ❛✈❛❧✐❛çã♦ ❞❛ ❡①♣r❡ssã♦ ❜♦♦❧❡❛♥❛ q✉❡ ❢♦r♠❛ ♦ ❝♦♥❞✐❝✐♦♥❛❧ ♦✉ ❧♦♦♣✳ P❛r❛ ❝♦♠❛♥❞♦s ❝♦♥❞✐❝✐♦♥❛✐s✱ ♦ ❡st❛❞♦ r❡t♦r♥❛❞♦ ♣❡❧❛ ❡①❡❝✉çã♦ ❞❡♣❡♥❞❡ ❞♦ r❡s✉❧✲ t❛❞♦ ❞❛ ❛✈❛❧✐❛çã♦ ❞❛ ❡①♣r❡ssã♦ ❜♦♦❧❡❛♥e✳ ❈❛s♦ ♦ r❡s✉❧t❛❞♦ s❡❥❛ ✈❡r❞❛❞❡✐r♦✱
♦ ❡st❛❞♦ r❡s✉❧t❛♥t❡ ❞❛ ❡①❡❝✉çã♦ ❞♦ ❝♦♠❛♥❞♦ ❛ss♦❝✐❛❞♦ ❛♦ t❤❡♥ é r❡t♦r♥❛❞♦✳ ❈❛s♦ ❝♦♥trár✐♦✱ ♦ ❡st❛❞♦ r❡s✉❧t❛♥t❡ ❞❛ ❛✈❛❧✐❛çã♦ ❞♦s ❝♦♠❛♥❞♦s ❛ss♦❝✐❛❞♦s ❛♦ ❜❧♦❝♦ ❡❧s❡ é r❡t♦r♥❛❞♦✳
❆s r❡❣r❛s ♣❛r❛ ❛✈❛❧✐❛çã♦ ❞❡ ❝♦♠❛♥❞♦s ❞❡ r❡♣❡t✐çã♦ sã♦ ❞❡✜♥✐❞❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ r❡s✉❧t❛❞♦ ❞❡ s❡ ❡①❡❝✉t❛r ❛ ❡①♣r❡ssã♦ ❜♦♦❧❡❛♥❛ e✳ ❈❛s♦ e❛✈❛❧✐❡ ♣❛r❛
❢❛❧s❡✱ ❛ r❡♣❡t✐çã♦ é ✜♥❛❧✐③❛❞❛ ❡ ♦ ❡st❛❞♦ r❡s✉❧t❛♥t❡ é ♦ ❡st❛❞♦ ♥❛ q✉❛❧ ❛ ❡①♣r❡ssã♦e❢♦✐ ❛✈❛❧✐❛❞❛✳ ❈❛s♦e❛✈❛❧✐❡ ♣❛r❛ ✈❡r❞❛❞❡✐r♦✱ t❡♠♦s q✉❡ ♦ ❝♦♠❛♥❞♦ cé ❡①❡❝✉t❛❞♦ ❡ ♦ ❡st❛❞♦ r❡t♦r♥❛❞♦ é ♦ r❡s✉❧t❛♥t❡ ❞❡ ✉♠❛ ♥♦✈❛ ❡①❡❝✉çã♦ ❞♦
❝♦♠❛♥❞♦ ✇❤✐❧❡✳ ❊st❡s ❢❛t♦s sã♦ ❡①♣r❡ss♦s ♥❛ ✜❣✉r❛ ✷✼✳
hc0, σ0i →σ1 hc1, σ1i →σ2
❊✈❛❧✲❙❡q
hc0 ❀ c1, σ0i →σ2
❋✐❣✉r❛ ✷✺✿ ❆✈❛❧✐❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦♠❛♥❞♦s he, σi →tr✉❡ hc0, σi →σ′
❊✈❛❧✲■❢✲❚r✉❡
h✐❢ e t❤❡♥ c0 ❡❧s❡ c1i →σ′
he, σi →❢❛❧s❡ hc1, σi →σ′
❊✈❛❧✲■❢✲❋❛❧s❡
h✐❢ e t❤❡♥ c0 ❡❧s❡ c1i →σ′
❋✐❣✉r❛ ✷✻✿ ❆✈❛❧✐❛♥❞♦ ✉♠ ❈♦♠❛♥❞♦ ❈♦♥❞✐❝✐♦♥❛❧
P❛r❛ ❡♥❝❡rr❛r ❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ s❡♠â♥t✐❝❛ ♦♣❡r❛❝✐♦♥❛❧ ❞❛ ❧✐♥❣✉❛❣❡♠ ■▼P✱ ❢❛❧t❛ ❞❡t❛❧❤❛r ❛s r❡❣r❛s ♣❛r❛ ♦s ❝♦♠❛♥❞♦s ♣r✐♥t ❡ r❡❛❞✳ ❆ s❡♠â♥t✐❝❛ ❞♦ ❝♦♠❛♥❞♦ ♣r✐♥t ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ s✐♠✐❧❛r à ❞♦ ❝♦♠❛♥❞♦ s❦✐♣✱ ❥á q✉❡ ♥ã♦ ❛❧t❡r❛ ♦ ❡st❛❞♦ ❛t✉❛❧✳ P♦r s✉❛ ✈❡③ ♦ ❝♦♠❛♥❞♦ r❡❛❞ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ s✐♠✐❧❛r à ❛tr✐❜✉✐çã♦ ♣♦r r❛③õ❡s ❡✈✐❞❡♥t❡s✳ ❆ s❡♠â♥t✐❝❛ ♣❛r❛ ❡st❡s ❝♦♠❛♥❞♦s é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✷✽✳
✹ ❊str✉t✉r❛ ❞♦ ❈ó❞✐❣♦
◆❡st❛ s❡çã♦ é ❞❡s❝r✐t❛ ❛ ❡str✉t✉r❛ ❞♦ ❝ó❞✐❣♦ ✐♥✐❝✐❛❧ ❢♦r♥❡❝✐❞♦ ❝♦♠ ❡st❡ ❡♥✉♥✲ ❝✐❛❞♦✳
✹✳✶ ❊str✉t✉r❛ ❞❡ ▼ó❞✉❧♦s
❖ ❝ó❞✐❣♦ ❞❛ ✐♠♣❧❡♠❡♥t❛çã♦ ✐♥✐❝✐❛❧ ❞♦ ✐♥t❡r♣r❡t❛❞♦r ❞❡ ■▼P é ❞✐✈✐❞✐❞♦ ♥❛s s❡❣✉✐♥t❡s ♣❛st❛s✿
• ■♥t❡r♣✿ ❊st❛ ♣❛st❛ ❝♦♥té♠ ♦ ❛rq✉✐✈♦ ■♠♣■♥t❡r♣✳❤s q✉❡ ♣♦ss✉✐ ❛s ❢✉♥çõ❡s ♣❛r❛ ❛ ❡①❡❝✉çã♦ ❞❡ ♣r♦❣r❛♠❛s ■▼P✳
• ▲❡①❡r✿ ❊st❛ ♣❛st❛ ❝♦♥té♠ ❞♦✐s ❛rq✉✐✈♦s✿ ❚♦❦❡♥❉❡❢✳❤s ❡ ■♠♣▲❡①❡r✳❤s✳ ❖ ❛rq✉✐✈♦ ❚♦❦❡♥❉❡❢✳❤s ♣♦ss✉✐ ❛ ❞❡✜♥✐çã♦ ❞♦s t♦❦❡♥s ❞❡ ■▼P✱ ❥á ♦ ❛rq✉✐✈♦ ■♠♣▲❡①❡r✳❤s ♣♦ss✉✐ ❛s ❢✉♥çõ❡s ♣❛r❛ r❡❛❧✐③❛r ❛ ❛♥á❧✐s❡ ❧é①✐❝❛ ❞❡ ♣r♦❣r❛♠❛s ■▼P✳
• ▲✐❜s✿ ❊st❛ ♣❛st❛ ❝♦♥té♠ ❛ ❜✐❜❧✐♦t❡❝❛ ❞❡ P❛rs✐♥❣ ✈✐st❛ ❡♠ ❝❧❛ss❡✳ • P❛rs❡r✿ ❊st❛ ♣❛st❛ ❝♦♥té♠ ♦ ❛rq✉✐✈♦ ■♠♣P❛rs❡r✳❤s q✉❡ ✐♠♣❧❡♠❡♥t❛ ♦
❛♥❛❧✐s❛❞♦r s✐♥tát✐❝♦ ❞❡ ■▼P✳
he, σi →❢❛❧s❡
❊✈❛❧✲❲❤✐❧❡✲❋❛❧s❡
h✇❤✐❧❡ e ❞♦ c, σi →σ
he, σi →tr✉❡ hc, σi →σ1 h✇❤✐❧❡ e ❞♦ c, σ1i →σ2
❊✈❛❧✲❲❤✐❧❡✲❚r✉❡
h✇❤✐❧❡ e ❞♦ c, σi →σ2
❋✐❣✉r❛ ✷✼✿ ❆✈❛❧✐❛♥❞♦ ✉♠ ❈♦♠❛♥❞♦ ❲❤✐❧❡ he, σi → v
❊✈❛❧✲Pr✐♥t
h♣r✐♥t e, σi → σ
❊✈❛❧✲❘❡❛❞
hr❡❛❞ v, σi → σ◦[v7→m]
❋✐❣✉r❛ ✷✽✿ ❆✈❛❧✐❛♥❞♦ ❈♦♠❛♥❞♦s ♣r✐♥t ❡ r❡❛❞
• ❚❝✿ ❊st❛ ♣❛st❛ ❝♦♥té♠ ♦ ❛rq✉✐✈♦ ■♠♣❚❝✳❤s q✉❡ ✐♠♣❧❡♠❡♥t❛ ♦ ✈❡r✐✜❝❛✲ ❞♦r ❞❡ t✐♣♦s ❞❡ ♣r♦❣r❛♠❛s ■▼P✳
❖ ❛rq✉✐✈♦ ▼❛✐♥✳❤s✱ ♥❛ ♣❛st❛ r❛✐③✱ é ♦ ♣r♦❣r❛♠❛ ♣r✐♥❝✐♣❛❧ ❞♦ ✐♥t❡r♣r❡t❛❞♦r ❡ é r❡s♣♦♥sá✈❡❧ ♣♦r ✉♥✐r ❝❛❞❛ ✉♠ ❞❡st❡s ❝♦♠♣♦♥❡♥t❡s ♣❛r❛ ❛ ❡①❡❝✉çã♦ ❞❡ ♣r♦❣r❛♠❛s✳
✹✳✷ ❚✐♣♦s ❞❡ ❞❛❞♦s ♣❛r❛ ♣r♦❣r❛♠❛s ■▼P
◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦s t✐♣♦s ❞❡ ❞❛❞♦s ✉t✐❧✐③❛❞♦s ♥♦ ✐♥t❡r♣r❡t❛❞♦r ❞❡ ■▼P✳ ■♥✐❝✐❛r❡♠♦s ❞❡s❝r❡✈❡♥❞♦ ♦ t✐♣♦ ❞❡ ❞❛❞♦s ❚♦❦❡♥✱ ♥❛ s❡q✉ê♥❝✐❛ ❛♣r❡s❡♥✲ t❛r❡♠♦s ❛ ár✈♦r❡ ❞❡ s✐♥t❛①❡ ❡ ✜♥❛❧♠❡♥t❡ ❛s ♠ô♥❛❞❛s ✉t✐❧✐③❛❞❛s ♥❛ ❡t❛♣❛ ❞❡ ✈❡r✐✜❝❛çã♦ ❞❡ t✐♣♦s ❡ ❡①❡❝✉çã♦ ❞❡ ❝ó❞✐❣♦✳
✹✳✷✳✶ ❖ ❚✐♣♦ ❞❡ ❞❛❞♦s ❚♦❦❡♥
❖ t✐♣♦ ❞❡ ❞❛❞♦s ❚♦❦❡♥ é ❞❡✜♥✐❞♦ ♥♦ ❛rq✉✐✈♦ ❚♦❦❡♥❉❡❢✳❤s ♥❛ ♣❛st❛ ▲❡①❡r✳ ❊st❡ t✐♣♦ ❞❡ ❞❛❞♦s✱ ♣♦ss✉✐ ✉♠ ❝♦♥str✉t♦r ♣❛r❛ ❝❛❞❛ t♦❦❡♥ ❞❡✜♥✐❞♦ ♥❛ ✜❣✉r❛ ✶✳ ❖ t✐♣♦ ❞❡ ❞❛❞♦s ❚♦❦❡♥ ♣♦ss✉✐ ❞❡✜♥✐çã♦ s✐♠♣❧❡s ❡ ❞✐r❡t❛✱ ❡st❛ é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✷✾✳ ❆♦ ❧❛❞♦ ❞❡ ❝❛❞❛ ❝♦♥str✉t♦r ❞❡st❡ t✐♣♦✱ é ❛♣r❡s❡♥t❛❞♦ s♦❜r❡ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❝♦♠❡♥tár✐♦✱ ❛ r❡s♣❡❝t✐✈❛ str✐♥❣ q✉❡ ❡st❡ r❡♣r❡s❡♥t❛✳
✹✳✷✳✷ ❖ ❚✐♣♦ ❞❛ ➪r✈♦r❡ ❞❡ ❙✐♥t❛①❡ ❆❜str❛t❛
❆ ❞❡✜♥✐çã♦ ❞♦s t✐♣♦s q✉❡ ❝♦♠♣õ❡ ❛ ár✈♦r❡ s✐♥tát✐❝❛ ❞❡ ♣r♦❣r❛♠❛s ■▼P ❡stá ❝♦♥t✐❞❛ ♥♦ ❛rq✉✐✈♦ ■♠♣❙②♥t❛①✳❤s ❧♦❝❛❧✐③❛❞♦ ♥❛ ♣❛st❛ ❙②♥t❛①✳
❖ ♣r✐♠❡✐r♦ t✐♣♦ ❞❛ ár✈♦r❡ ❞❡ s✐♥t❛①❡ ❛ s❡r ❝♦♥s✐❞❡r❛❞♦ é ♦ q✉❡ ❞❡✜♥❡ ❛s ❡①♣r❡ssõ❡s ❛r✐t♠ét✐❝❛s✳ ❆ ❞❡✜♥✐çã♦ ❞❡st❡ é ♠♦str❛❞❛ ♥❛ ✜❣✉r❛ ✸✵✳ ❖s
❝♦♥str✉t♦r❡s ❞❡st❡ t✐♣♦ sã♦ ❛✉t♦✲❡①♣❧✐❝❛t✐✈♦s✳ ❚❡♠♦s ✉♠ ❝♦♥str✉t♦r ♣❛r❛ ❝❛❞❛ ✉♠ ❞❛s ❡①♣r❡ssõ❡s ❛r✐t♠ét✐❝❛s✱ ✉♠ ♣❛r❛ ❝♦♥st❛♥t❡s ✐♥t❡✐r❛s ❡ ✉♠ ♣❛r❛ ✈❛r✐á✈❡✐s✳
❊①♣r❡ssõ❡s ❡♥✈♦❧✈❡♥❞♦ ♦♣❡r❛❞♦r❡s ❜♦♦❧❡❛♥♦s sã♦ r❡♣r❡s❡♥t❛❞❛s ✉t✐❧✐③❛♥❞♦ ♦ t✐♣♦ ❞❡✜♥✐❞♦ ♥❛ ✜❣✉r❛ ✸✶✳ ❆ss✐♠ ❝♦♠♦ ♦ t✐♣♦ ❞❡ ❡①♣r❡ssõ❡s ❛r✐t♠ét✐❝❛s✱ ♦ ❞❡ ❡①♣r❡ssõ❡s ❜♦♦❧❡❛♥❛s ♣♦ss✉✐ ✉♠ ❝♦♥str✉t♦r ♣❛r❛ ❝❛❞❛ ✉♠ ❞♦s ♦♣❡r❛❞♦r❡s ❧ó❣✐❝♦s ✭❡✱ ♦✉✱ ♥❡❣❛çã♦✮✱ ♦♣❡r❛❞♦r❡s r❡❧❛❝✐♦♥❛✐s ✭✐❣✉❛❧❞❛❞❡ ❡ ❝♦♠♣❛r❛çã♦✮✱ ❝♦♥st❛♥t❡s ❜♦♦❧❡❛♥❛s✱ ❝♦♥st❛♥t❡s str✐♥❣ ❡ ❡①♣r❡ssõ❡s ❛r✐t♠ét✐❝❛s✳
❖ t✐♣♦ ❞❡ ❞❛❞♦s ♣❛r❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❝♦♠❛♥❞♦s ■▼P é ❛♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛ ✸✷ ❡ ♣♦ss✉✐ s✐❣♥✐✜❝❛❞♦ ✐♥t✉✐t✐✈♦✳
✹✳✷✳✸ ❆ ▼ô♥❛❞❛ ❞❡ ❱❡r✐✜❝❛çã♦ ❞❡ ❚✐♣♦s
P❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ✈❡r✐✜❝❛çã♦ ❞❡ t✐♣♦s ❞❡✈❡♠♦s ♠❛♥t❡r ✉♠ ❝♦♥t❡①t♦ ❞❡ t✐♣♦s ♣❛r❛ ❛r♠❛③❡♥❛r ♦ t✐♣♦ ❞❡ ❝❛❞❛ ✈❛r✐á✈❡❧ ❞❡✜♥✐❞❛ ♥♦ ♣r♦❣r❛♠❛✳ ❖ ♠ó❞✉❧♦ ■♠♣❚❝✳❤s✱ ♣r❡s❡♥t❡ ♥❛ ♣❛st❛ ❚❝✱ ❞❡✜♥❡ ✉♠❛ ♠ô♥❛❞❛ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ✈❡r✐✜❝❛çã♦ ❞❡ t✐♣♦s ❞❡ ♣r♦❣r❛♠❛s ■▼P✳
❯♠ ❝♦♥t❡①t♦ ❞❡ t✐♣♦s é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ▼❛♣ ✭♠ó❞✉❧♦ ❉❛t❛✳▼❛♣✮ ❝✉❥❛ ❝❤❛✈❡ sã♦ ✈❛r✐á✈❡✐s ❡ ♦s ✈❛❧♦r❡s sã♦ t✐♣♦s ✭✜❣✉r❛ ✸✸✮✳
P❛r❛ r❡♣r❡s❡♥t❛r ❛ ♠ô♥❛❞❛ ✉t✐❧✐③❛r❡♠♦s ♦ tr❛♥s❢♦r♠❛❞♦r ♠♦♥á❞✐❝♦ ❙t❛t❡❚ ✭♠ó❞✉❧♦ ❈♦♥tr♦❧✳▼♦♥❛❞✳❙t❛t❡✮✳ ❆ ❞❡✜♥✐çã♦ ❞❡st❡ é ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✸✹✳
✹✳✷✳✹ ❆ ▼ô♥❛❞❛ ❞❡ ❊①❡❝✉çã♦ ❞❡ ❈ó❞✐❣♦
P❛r❛ ❡①❡❝✉t❛r ♣r♦❣r❛♠❛s ■▼P ❞❡✈❡♠♦s ♠❛♥t❡r ✉♠ ❛♠❜✐❡♥t❡ q✉❡ ❛r♠❛③❡♥❛ ♦s ✈❛❧♦r❡s ❛t✉❛✐s ❞❡ ❝❛❞❛ ✈❛r✐á✈❡❧ ❞❡✜♥✐❞❛ ♥♦ ♣r♦❣r❛♠❛✳ ❊st❡ ❛♠❜✐❡♥t❡ r❡♣r❡s❡♥t❛ ❛ ❢✉♥çã♦σ ❞❡✜♥✐❞❛ ♥❛ s❡çã♦ ✸✳✹✳ ❯♠❛ ✈❡③ q✉❡ ❛ ❢✉♥çã♦ σ r❡❛❧✐③❛
♦ ♠❛♣❡❛♠❡♥t♦ ❡♥tr❡ ✈❛r✐á✈❡✐s ❡ ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛ ❡♥tr❡ ♦s t✐♣♦s ✐♥t ❡ ❜♦♦❧✱ ✉s❛r❡♠♦s ♦ t✐♣♦ ❞❡ ❞❛❞♦s ❍❛s❦❡❧❧ ❊✐t❤❡r ❛ ❜ q✉❡ r❡♣r❡s❡♥t❛ ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛ ❡♥tr❡ ♦s t✐♣♦s ❛ ❡ ❜✳ ❙❡♥❞♦ ❛ss✐♠ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❛ ❢✉♥çã♦ σ ❝♦♠♦
s❡♥❞♦ ✉♠ ▼❛♣ ❝✉❥❛s ❝❤❛✈❡s sã♦ ✈❛r✐á✈❡✐s ❡ ♦s ✈❛❧♦r❡s sã♦ ✈❛❧♦r❡s ✐♥t❡✐r♦s ♦✉ ❜♦♦❧❡❛♥♦s✱ ✐st♦ é ❊✐t❤❡r ■♥t ❇♦♦❧✳
❆ ♠ô♥❛❞❛ ♣❛r❛ ❡①❡❝✉çã♦ é ❞❡✜♥✐❞❛ ♥❛ ✜❣✉r❛ ✸✻✳ ❙✉❛ ❞❡✜♥✐çã♦ é ✐❞ê♥t✐❝❛ à ❞❡ ✈❡r✐✜❝❛çã♦ ❞❡ t✐♣♦s✳
✹✳✸ ❋✉♥çõ❡s ❛ s❡r ■♠♣❧❡♠❡♥t❛❞❛s
◆❡st❛ s❡çã♦ é ❞❡✜♥✐❞♦ ♦ q✉❡ ❞❡✈❡rá s❡r ✐♠♣❧❡♠❡♥t❛❞♦ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞♦ tr❛❜❛❧❤♦✳
✹✳✸✳✶ ▼❡❧❤♦r❛r ❛ ■♥t❡r❢❛❝❡ ❝♦♠ ♦ ❯s✉ár✐♦ ❞♦ ■♥t❡r♣r❡t❛❞♦r ✖ ❱❛❧♦r ✵✱✷✺
❆ ❢✉♥çã♦ ♣r✐♥❝✐♣❛❧ ❞♦ ✐♥t❡r♣r❡t❛❞♦r ❞❡ ■▼P ❡stá ❧♦❝❛❧✐③❛❞❛ ♥❛ ♣❛st❛ r❛✐③ ❞♦ ❝ó❞✐❣♦ ❢♦r♥❡❝✐❞♦ ❝♦♠ ♦ ♥♦♠❡ ▼❛✐♥✳❤s ✳ ❆té ♦ ♠♦♠❡♥t♦✱ ♦ ✐♥t❡r♣r❡t❛❞♦r ❛❝❡✐t❛✱ ❡♠ ❧✐♥❤❛ ❞❡ ❝♦♠❛♥❞♦✱ ❛♣❡♥❛s ✉♠ ♣❛râ♠❡tr♦✿ ♦ ♥♦♠❡ ❞♦ ❛rq✉✐✈♦ ❞♦ ♣r♦❣r❛♠❛ ❛ s❡r ❡①❡❝✉t❛❞♦✳ ▼♦❞✐✜q✉❡ ♦ ❝ó❞✐❣♦ ❞❡st❡ ❛rq✉✐✈♦ ❞❡ ♠♦❞♦ q✉❡ ♦ ✐♥t❡r♣r❡t❛❞♦r ❛❝❡✐t❡ ❛s s❡❣✉✐♥t❡s ♦♣çõ❡s✿
• ✲❧✿ Pár❛ ♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞♦ ♣r♦❣r❛♠❛ ❢♦r♥❡❝✐❞♦ ❛♣ós ❛ ❛♥á❧✐s❡ ❧é①✐❝❛ ❡ ✐♠♣r✐♠❡ t♦❞♦s ♦s t♦❦❡♥s ♣r♦❝❡ss❛❞♦s✳
• ✲♣✿ Pár❛ ♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞♦ ♣r♦❣r❛♠❛ ❢♦r♥❡❝✐❞♦ ❛♣ós ❛ ❛♥á❧✐s❡ s✐♥tá✲ t✐❝❛ ❡ ✐♠♣r✐♠❡ ❛ ár✈♦r❡ ❞❡ s✐♥t❛①❡ ❞♦ ♣r♦❣r❛♠❛✳
✹✳✸✳✷ ■♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❆♥❛❧✐s❛❞♦r ▲é①✐❝♦ ✖ ❱❛❧♦r ✵✱✷✺
■♠♣❧❡♠❡♥t❡ ♦ ❛♥❛❧✐s❛❞♦r ❧é①✐❝♦ ❞❡ ■▼P ♥♦ ❛rq✉✐✈♦ ■♠♣▲❡①❡r✳❤s✳ P❛r❛ ✐ss♦ ✈♦❝ê ❞❡✈❡rá ✐♠♣❧❡♠❡♥t❛r ❛ ❢✉♥çã♦✿
❧❡①❡r ✿✿ ❙tr✐♥❣ ✲❃ ❬❚♦❦❡♥❪
q✉❡ ❛ ♣❛rt✐r ❞❛ ❙tr✐♥❣ ❞❡ ❡♥tr❛❞❛ ❞♦ ♣r♦❣r❛♠❛ ❢♦♥t❡ ♣r♦❞✉③✐rá ✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s t♦❦❡♥s ♣r❡s❡♥t❡s ♥❡st❛✳
✹✳✸✳✸ ■♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❆♥❛❧✐s❛❞♦r ❙✐♥tát✐❝♦ ✖ ❱❛❧♦r ✵✱✷✺ ■♠♣❧❡♠❡♥t❡ ♦ ❛♥❛❧✐s❛❞♦r s✐♥tát✐❝♦ ❞❡ ■▼P ♥♦ ❛rq✉✐✈♦ ■♠♣P❛rs❡r✳❤s✳ P❛r❛ ✐ss♦ ✈♦❝ê ❞❡✈❡rá ✐♠♣❧❡♠❡♥t❛r ❛ ❢✉♥çã♦✿
♣❛rs❡r ✿✿ ❬❚♦❦❡♥❪ ✲❃ Pr♦❣r❛♠
❡st❛ ❢✉♥çã♦ ❞❡✈❡rá r❡❝♦♥❤❡❝❡r ♣r♦❣r❛♠❛s ■▼P ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❣r❛♠át✐❝❛ ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✷✳
✹✳✸✳✹ ■♠♣❧❡♠❡♥t❛r ■♥stâ♥❝✐❛s ❞❡ ❙❤♦✇ ♣❛r❛ ❛ ➪r✈♦r❡ ❞❡ ❙✐♥t❛①❡ ❆❜str❛t❛ ✖ ❱❛❧♦r ✵✱✷✺
❆❧é♠ ❞✐ss♦✱ ♥♦ ❛rq✉✐✈♦ ■♠♣❙②♥t❛①✳❤s✱ ✈♦❝ê ❞❡✈❡rá ✐♠♣❧❡♠❡♥t❛r ✐♥stâ♥❝✐❛s ❞❛ ❝❧❛ss❡ ❙❤♦✇ ♣❛r❛ q✉❡ ♦ r❡s✉❧t❛❞♦ ❞❡ ✐♠♣r✐♠✐r ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❛ s✐♥t❛①❡ ❞❡ ■▼P s❡❥❛ s✐♠✐❧❛r ❛♦ t❡①t♦ ❞♦ ♣r♦❣r❛♠❛ ♣r♦❝❡ss❛❞♦ ♣❡❧♦ ✐♥t❡r♣r❡t❛❞♦r✳ ✹✳✸✳✺ ■♠♣❧❡♠❡♥t❛çã♦ ❞❛ ❱❡r✐✜❝❛çã♦ ❞❡ ❚✐♣♦s ✖ ❱❛❧♦r ✵✱✷✺ ■♠♣❧❡♠❡♥t❡ ♦ ❛❧❣♦r✐t♠♦ ♣❛r❛ ✈❡r✐✜❝❛çã♦ ❞❡ t✐♣♦s ♣❛r❛ ❛ ❧✐♥❣✉❛❣❡♠ ■▼P ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s r❡❣r❛s ❛♣r❡s❡♥t❛❞❛s ♥❛ s❡çã♦ ✸✳✸✳ ❖ ❛❧❣♦r✐t♠♦ ❞❡✈❡ s❡r ✐♠♣❧❡♠❡♥t❛❞♦ ♣❡❧❛ ❢✉♥çã♦
t❝ ✿✿ Pr♦❣r❛♠ ✲❃ ■❖ ✭✮ ♥♦ ♠ó❞✉❧♦ ■♠♣❚❝✳❤s ♥❛ ♣❛st❛ ❚❝✳
✹✳✸✳✻ ■♠♣❧❡♠❡♥t❛çã♦ ❞♦ ■♥t❡r♣r❡t❛❞♦r ❞❡ ■▼P ✖ ❱❛❧♦r ✵✱✷✺ ■♠♣❧❡♠❡♥t❡ ♦ ❛❧❣♦r✐t♠♦ ♣❛r❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ♣r♦❣r❛♠❛s ■▼P ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s r❡❣r❛s ❞❡ s❡♠â♥t✐❝❛ ♦♣❡r❛❝✐♦♥❛❧ ❛♣r❡s❡♥t❛❞❛s ♥❛ s❡çã♦ ✸✳✹✳ ❊st❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡✈❡rá ❡st❛r ❝♦♥t✐❞❛ ♥♦ ♠ó❞✉❧♦ ■♠♣■♥t❡r♣✳❤s ❧♦❝❛❧✐③❛❞♦ ♥❛ ♣❛st❛ ■♥t❡r♣✳
✺ ❉✐❝❛s
• ◆ã♦ ❞❡✐①❡ ♣❛r❛ ❛ ú❧t✐♠❛ ❤♦r❛ ♦ ✐♥í❝✐♦ ❞❛ s♦❧✉çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❊❧❡ ❡♥✈♦❧✈❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ s✉❜st❛♥❝✐❛❧ ❞❡ ❝ó❞✐❣♦✳
• ➱ r❡❝♦♠❡♥❞á✈❡❧ q✉❡ ✈♦❝ê ❞✐✈✐❞❛ ♦ s❡✉ ❝ó❞✐❣♦ ❡♠ ❞✐✈❡rs❛s ♣❡q✉❡♥❛s ❢✉♥çõ❡s ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ t❛r❡❢❛ ❞❡ t❡st❡s✳
• ◆ã♦ ✐♥✐❝✐❡ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❡t❛♣❛ ❞♦ ✐♥t❡r♣r❡t❛✲ ❞♦r s❡♠ ❛ ❝♦♥❝❧✉sã♦ ❞❛ ❡t❛♣❛ ❛♥t❡r✐♦r✳
• ❈r✐❡ ❞✐✈❡rs♦s ♣r♦❣r❛♠❛s ♣❛r❛ t❡st❛r s❡✉ ✐♥t❡r♣r❡t❛❞♦r✳ ❆♦ ❡♥✈✐❛r s❡✉ ❝ó❞✐❣♦ ❛♥❡①❡ t♦❞♦s ♦s t❡st❡s ✉t✐❧✐③❛❞♦s ♣♦r ✈♦❝ê✳
✻ ❘❡q✉✐s✐t♦s ❞♦ ❚r❛❜❛❧❤♦
• ❙❡✉ ♣r♦❣r❛♠❛ ❞❡✈❡ ❡①❡❝✉t❛r s♦❜r❡ ❛ ú❧t✐♠❛ ✈❡rsã♦ ❞❛ ♣❧❛t❛❢♦r♠❛ ❍❛s✲ ❦❡❧❧✳
• ❚♦❞❛ ❛ ❞♦❝✉♠❡♥t❛çã♦ ♣r♦❞✉③✐❞❛ ❞❡✈❡ ❡st❛r ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ♥♦r♠❛s ♣r❡s❡♥t❡s ♥♦ sít✐♦ ❞❡st❛ ❞✐s❝✐♣❧✐♥❛✳
• ❊st❡ tr❛❜❛❧❤♦ ♣♦❞❡ s❡r ❢❡✐t♦ ♣♦r ❣r✉♣♦s ❞❡ ❛té ✸ ❛❧✉♥♦s✳ P♦rt❛♥t♦✱ ♥ã♦ ❤❛✈❡rá✱ s♦❜ ♥❡♥❤✉♠❛ ❤✐♣ót❡s❡✱ ❣r✉♣♦s ❝♦♠ ♠❛✐s ❞❡ ✸ ❛❧✉♥♦s✳
• ◗✉❛❧q✉❡r ♣❧á❣✐♦ ✐♠♣❧✐❝❛rá ❡♠ ❩❊❘❖ ♥❛ ♥♦t❛ ❞❡ ❚❖❉❖❙ ♦s tr❛❜❛❧❤♦s ♣rát✐❝♦s ♣❛r❛ t♦❞♦s ♦s ❛❧✉♥♦s ❡♥✈♦❧✈✐❞♦s✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ❡✈✐t❛r ❛ ♣rát✐❝❛ ❞♦ ♣❧á❣✐♦✱ ♦s ❛❧✉♥♦s ❡♥✈♦❧✈✐❞♦s t❡rã♦ ❛ ♠❛✐♦r ♥♦t❛ ❞❛s três ♣r♦✈❛s ❛♥✉❧❛❞❛ ✭③❡r❛❞❛✮✳
• ❙♦♠❡♥t❡ r❡❝❡❜❡rá ♥♦t❛ ❞♦ tr❛❜❛❧❤♦ ♦ ❛❧✉♥♦ q✉❡ ♦❜t❡r ♣❡❧♦ ♠❡♥♦s ✸✵✪ ❞❛ ♥♦t❛ ♥❛ ♣r♦✈❛ ✸✳
• P♦❞❡rã♦ s❡r ♠❛r❝❛❞❛s ❡♥tr❡✈✐st❛s ❛ ❝r✐tér✐♦ ❞♦ ♣r♦❢❡ss♦r✳ • ❉❛t❛ ❞❡ ❊♥tr❡❣❛✿ ✶✶✴✵✼✴✷✵✶✶
❞❛t❛ ❚♦❦❡♥ ❂ ◆❛♠❡ ❙tr✐♥❣ ✲✲ ✈❛r✐❛❜❧❡
⑤ ◆✉♠❜❡r ■♥t ✲✲ ✐♥t❡❣❡r ❝♦♥st❛♥t
⑤ ❚♣❧✉s ✲✲ ✬✰✬
⑤ ❚♣r✐♥t ✲✲ ♣r✐♥t
⑤ ❚r❡❛❞ ✲✲ r❡❛❞
⑤ ❚♠✐♥✉s ✲✲ ✬✲✬
⑤ ❚♠✉❧t ✲✲ ✬✯✬
⑤ ❚❞✐✈ ✲✲ ✬✴✬
⑤ ❚❡q ✲✲ ✬❂✬
⑤ ❚❧❡q ✲✲ ✬❁❂✬
⑤ ❚♥♦t ✲✲ ✬✦✬
⑤ ❚❛♥❞ ✲✲ ✬✫✬
⑤ ❚♦r ✲✲ ✬⑤✬
⑤ ❚s❦✐♣ ✲✲ s❦✐♣
⑤ ❚❛ss✐❣♥ ✲✲ ✬✿❂✬
⑤ ❚s❡♠✐ ✲✲ ✬❀✬
⑤ ❚✐❢ ✲✲ ✐❢
⑤ ❚t❤❡♥ ✲✲ t❤❡♥
⑤ ❚❡❧s❡ ✲✲ ❡❧s❡
⑤ ❚✇❤✐❧❡ ✲✲ ✇❤✐❧❡
⑤ ❚❞♦ ✲✲ ❞♦
⑤ ❚▲❇r❛❝❡ ✲✲ ✬④✬
⑤ ❚❘❇r❛❝❡ ✲✲ ✬⑥✬
⑤ ❚▲P❛r❡♥ ✲✲ ✬✭✬
⑤ ❚❘P❛r❡♥ ✲✲ ✬✮✬
⑤ ❚tr✉❡ ✲✲ tr✉❡
⑤ ❚❢❛❧s❡ ✲✲ ❢❛❧s❡
⑤ ❚♣r♦❣r❛♠ ✲✲ ♣r♦❣r❛♠
⑤ ❚✈❛r ✲✲ ✈❛r
⑤ ❚✐♥t ✲✲ ✐♥t
⑤ ❚❜♦♦❧ ✲✲ ❜♦♦❧
⑤ ❚str✐♥❣ ❙tr✐♥❣ ✲✲ str✐♥❣ ❝♦♥st❛♥t ❞❡r✐✈✐♥❣ ✭❖r❞✱ ❙❤♦✇✮
❋✐❣✉r❛ ✷✾✿ ❚✐♣♦ ❞❡ ❉❛❞♦s ❚♦❦❡♥
❞❛t❛ ❆❊①♣ ❂ ❈♦♥st ■♥t ✲✲ ❛ ✐♥t❡❣❡r ❝♦♥st❛♥t ⑤ ❱❛r ❱❛r✐❛❜❧❡ ✲✲ ❛ ✈❛r✐❛❜❧❡
⑤ ❆❊①♣ ✿✰✿ ❆❊①♣ ✲✲ s✉♠♠✐♥❣ t✇♦ ❡①♣r❡ss✐♦♥s ⑤ ❆❊①♣ ✿✲✿ ❆❊①♣ ✲✲ s✉❜tr❛❝t✐♥❣ t✇♦ ❡①♣r❡ss✐♦♥s ⑤ ❆❊①♣ ✿✯✿ ❆❊①♣ ✲✲ ♠✉❧t✐♣❧②✐♥❣ t✇♦ ❡①♣r❡ss✐♦♥s ⑤ ❆❊①♣ ✿✴✿ ❆❊①♣ ✲✲ ❞✐✈✐❞✐♥❣ t✇♦ ❡①♣r❡ss✐♦♥s ❞❡r✐✈✐♥❣ ✭❊q✱ ❖r❞✮
❋✐❣✉r❛ ✸✵✿ ❚✐♣♦ ❞❡ ❉❛❞♦s ♣❛r❛ ❊①♣r❡ssõ❡s ❆r✐t♠ét✐❝❛s ❞❛t❛ ❇❊①♣ ❂ ❇❈♦♥st ❇♦♦❧ ✲✲ ❜♦♦❧❡❛♥ ❝♦♥st❛♥ts
⑤ ❇❊①♣ ✿❂✿ ❇❊①♣ ✲✲ ❛r✐t❤♠❡t✐❝ ❡q✉❛❧✐t②
⑤ ❇❊①♣ ✿❁❂✿ ❇❊①♣ ✲✲ ❛r✐t❤♠❡t✐❝ ❧❡ss✲t❤❛♥✲❡q✉❛❧ ⑤ ◆♦t ❇❊①♣ ✲✲ ♥❡❣❛t✐♦♥
⑤ ❇❊①♣ ✿✫✿ ❇❊①♣ ✲✲ ❝♦♥❥✉♥❝t✐♦♥ ⑤ ❇❊①♣ ✿⑤✿ ❇❊①♣ ✲✲ ❞✐s❥✉♥❝t✐♦♥
⑤ ❇◆✉♠ ❆❊①♣ ✲✲ ❛r✐t❤♠❡t✐❝ ❡①♣r❡ss✐♦♥ ⑤ ❇❙tr ❙tr✐♥❣ ✲✲ str✐♥❣ ❧✐t❡r❛❧
❞❡r✐✈✐♥❣ ✭❊q✱ ❖r❞✮
❋✐❣✉r❛ ✸✶✿ ❚✐♣♦ ❞❡ ❉❛❞♦s ♣❛r❛ ❊①♣r❡ssõ❡s ❇♦♦❧❡❛♥❛s
❞❛t❛ ❈♦♠ ❂ ❙❦✐♣ ✲✲ ❞♦ ♥♦t❤✐♥❣ ❝♦♠♠❛♥❞
⑤ Pr✐♥t ❇❊①♣ ✲✲ ♣r✐♥t ❝♦♠♠❛♥❞
⑤ ❘❡❛❞ ❱❛r✐❛❜❧❡ ✲✲ r❡❛❞ ❝♦♠♠❛♥❞ ⑤ ❆ss✐❣♥ ❱❛r✐❛❜❧❡ ❇❊①♣ ✲✲ ❛ss✐❣♥♠❡♥t
⑤ ■❢ ❇❊①♣ ❬❈♦♠❪ ❬❈♦♠❪ ✲✲ ❝♦♥❞✐t✐♦♥❛❧ ❡①❡❝✉t✐♦♥ ⑤ ❲❤✐❧❡ ❇❊①♣ ❬❈♦♠❪ ✲✲ ✇❤✐❧❡ ❧♦♦♣
❞❡r✐✈✐♥❣ ✭❊q✱ ❖r❞✮
❋✐❣✉r❛ ✸✷✿ ❚✐♣♦ ❞❡ ❉❛❞♦s ♣❛r❛ ❈♦♠❛♥❞♦s t②♣❡ ❈♦♥t❡①t ❂ ▼❛♣✳▼❛♣ ❱❛r✐❛❜❧❡ ❚②
❋✐❣✉r❛ ✸✸✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❝♦♥t❡①t♦s ❞❡ t✐♣♦s t②♣❡ ❚❝▼ ❛ ❂ ✭❙t❛t❡❚ ❈♦♥t❡①t ■❖✮ ❛
❋✐❣✉r❛ ✸✹✿ ❉❡✜♥✐çã♦ ❞❛ ▼ô♥❛❞❛ ❞❡ ✈❡r✐✜❝❛çã♦ ❞❡ t✐♣♦s t②♣❡ ❱❛r❊♥✈ ❂ ▼❛♣✳▼❛♣ ❱❛r✐❛❜❧❡ ✭❊✐t❤❡r ■♥t ❇♦♦❧✮
❋✐❣✉r❛ ✸✺✿ ❉❡✜♥✐çã♦ ❞♦ ❛♠❜✐❡♥t❡ ♣❛r❛ ❡①❡❝✉çã♦
t②♣❡ ❊①❡❝▼ ❛ ❂ ✭❙t❛t❡❚ ❱❛r❊♥✈ ■❖✮ ❛
❋✐❣✉r❛ ✸✻✿ ❉❡✜♥✐çã♦ ❞❛ ♠ô♥❛❞❛ ♣❛r❛ ❡①❡❝✉çã♦
