rs r r - - PowerPoint PPT Presentation
rs r r tr trts rr r rr t r t
❣❡r❛r❞♦❅✐✜✳✉✐♦✳♥♦
❉❡♣❛rt♠❡♥t ♦❢ ■♥❢♦r♠❛t✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❖s❧♦
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶ ✴ ✹✷
❚❤✐s ❞❡❡❞ ♦❢ ❆❣r❡❡♠❡♥t ✐s ♠❛❞❡ ❜❡t✇❡❡♥✿ ✶✳ ❬♥❛♠❡❪✱ ❢r♦♠ ♥♦✇ ♦♥ r❡❢❡rr❡❞ t♦ ❛s Pr♦✈✐❞❡r ❛♥❞ ✷✳ t❤❡ ❈❧✐❡♥t✳ ■◆❚❘❖❉❯❈❚■❖◆ ✸✳ ❚❤❡ Pr♦✈✐❞❡r ✐s ♦❜❧✐❣❡❞ t♦ ♣r♦✈✐❞❡ t❤❡ ■♥t❡r♥❡t ❙❡r✈✐❝❡s ❛s st✐♣✉❧❛t❡❞ ✐♥ t❤✐s ❆❣r❡❡♠❡♥t✳ ✹✳ ❉❊❋■◆■❚■❖◆❙ ❛✮ ■♥t❡r♥❡t tr❛✣❝ ♠❛② ❜❡ ♠❡❛s✉r❡❞ ❜② ❜♦t❤ ❈❧✐❡♥t ❛♥❞ Pr♦✈✐❞❡r ❜② ♠❡❛♥s ♦❢ ❊q✉✐♣♠❡♥t ❛♥❞ ♠❛② t❛❦❡ t❤❡ t✇♦ ✈❛❧✉❡s ❤✐❣❤ ❛♥❞ ♥♦r♠❛❧✳ ❖P❊❘❆❚■❱❊ P❆❘❚ ✶✳ ❚❤❡ ❈❧✐❡♥t s❤❛❧❧ ♥♦t s✉♣♣❧② ❢❛❧s❡ ✐♥❢♦r♠❛t✐♦♥ t♦ t❤❡ ❈❧✐❡♥t ❘❡❧❛t✐♦♥s ❉❡♣❛rt♠❡♥t ♦❢ t❤❡ Pr♦✈✐❞❡r✳ ✷✳ ❲❤❡♥❡✈❡r t❤❡ ■♥t❡r♥❡t ❚r❛✣❝ ✐s ❤✐❣❤ t❤❡♥ t❤❡ ❈❧✐❡♥t ♠✉st ♣❛② ♣r✐❝❡ ✐♠♠❡❞✐❛t❡❧②✱ ♦r t❤❡ ❈❧✐❡♥t ♠✉st ♥♦t✐❢② t❤❡ Pr♦✈✐❞❡r ❜② s❡♥❞✐♥❣ ❛♥ ❡✲♠❛✐❧ s♣❡❝✐❢②✐♥❣ t❤❛t ❤❡ ✇✐❧❧ ♣❛② ❧❛t❡r✳ ✸✳ ■❢ t❤❡ ❈❧✐❡♥t ❞❡❧❛②s t❤❡ ♣❛②♠❡♥t ❛s st✐♣✉❧❛t❡❞ ✐♥ ✷✱ ❛❢t❡r ♥♦t✐✜❝❛t✐♦♥ ❤❡ ♠✉st ✐♠♠❡❞✐❛t❡❧② ❧♦✇❡r t❤❡ ■♥t❡r♥❡t tr❛✣❝ t♦ t❤❡ ♥♦r♠❛❧ ❧❡✈❡❧✱ ❛♥❞ ♣❛② ❧❛t❡r t✇✐❝❡ ✭✷ ♣r✐❝❡ ✮✳ ✹✳ ■❢ t❤❡ ❈❧✐❡♥t ❞♦❡s ♥♦t ❧♦✇❡r t❤❡ ■♥t❡r♥❡t tr❛✣❝ ✐♠♠❡❞✐❛t❡❧②✱ t❤❡♥ t❤❡ ❈❧✐❡♥t ✇✐❧❧ ❤❛✈❡ t♦ ♣❛② ✸ ♣r✐❝❡ ✳ ✺✳ ❚❤❡ ❈❧✐❡♥t s❤❛❧❧✱ ❛s s♦♦♥ ❛s t❤❡ ■♥t❡r♥❡t ❙❡r✈✐❝❡ ❜❡❝♦♠❡s ♦♣❡r❛t✐✈❡✱ s✉❜♠✐t ✇✐t❤✐♥ s❡✈❡♥ ✭✼✮ ❞❛②s t❤❡ P❡rs♦♥❛❧ ❉❛t❛ ❋♦r♠ ❢r♦♠ ❤✐s ❛❝❝♦✉♥t ♦♥ t❤❡ Pr♦✈✐❞❡r✬s ✇❡❜ ♣❛❣❡ t♦ t❤❡ ❈❧✐❡♥t ❘❡❧❛t✐♦♥s ❉❡♣❛rt♠❡♥t ♦❢ t❤❡ Pr♦✈✐❞❡r✳
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷ ✴ ✹✷
❚❤✐s ❞❡❡❞ ♦❢ ❆❣r❡❡♠❡♥t ✐s ♠❛❞❡ ❜❡t✇❡❡♥✿ ✶✳ ❬♥❛♠❡❪✱ ❢r♦♠ ♥♦✇ ♦♥ r❡❢❡rr❡❞ t♦ ❛s Pr♦✈✐❞❡r ❛♥❞ ✷✳ t❤❡ ❈❧✐❡♥t✳ ■◆❚❘❖❉❯❈❚■❖◆ ✸✳ ❚❤❡ Pr♦✈✐❞❡r ✐s ♦❜❧✐❣❡❞ t♦ ♣r♦✈✐❞❡ t❤❡ ■♥t❡r♥❡t ❙❡r✈✐❝❡s ❛s st✐♣✉❧❛t❡❞ ✐♥ t❤✐s ❆❣r❡❡♠❡♥t✳ ✹✳ ❉❊❋■◆■❚■❖◆❙ ❛✮ ■♥t❡r♥❡t tr❛✣❝ ♠❛② ❜❡ ♠❡❛s✉r❡❞ ❜② ❜♦t❤ ❈❧✐❡♥t ❛♥❞ Pr♦✈✐❞❡r ❜② ♠❡❛♥s ♦❢ ❊q✉✐♣♠❡♥t ❛♥❞ ♠❛② t❛❦❡ t❤❡ t✇♦ ✈❛❧✉❡s ❤✐❣❤ ❛♥❞ ♥♦r♠❛❧✳ ❖P❊❘❆❚■❱❊ P❆❘❚ ✶✳ ❚❤❡ ❈❧✐❡♥t s❤❛❧❧ ♥♦t s✉♣♣❧② ❢❛❧s❡ ✐♥❢♦r♠❛t✐♦♥ t♦ t❤❡ ❈❧✐❡♥t ❘❡❧❛t✐♦♥s ❉❡♣❛rt♠❡♥t ♦❢ t❤❡ Pr♦✈✐❞❡r✳ ✷✳ ❲❤❡♥❡✈❡r t❤❡ ■♥t❡r♥❡t ❚r❛✣❝ ✐s ❤✐❣❤ t❤❡♥ t❤❡ ❈❧✐❡♥t ♠✉st ♣❛② [♣r✐❝❡] ✐♠♠❡❞✐❛t❡❧②✱ ♦r t❤❡ ❈❧✐❡♥t ♠✉st ♥♦t✐❢② t❤❡ Pr♦✈✐❞❡r ❜② s❡♥❞✐♥❣ ❛♥ ❡✲♠❛✐❧ s♣❡❝✐❢②✐♥❣ t❤❛t ❤❡ ✇✐❧❧ ♣❛② ❧❛t❡r✳ ✸✳ ■❢ t❤❡ ❈❧✐❡♥t ❞❡❧❛②s t❤❡ ♣❛②♠❡♥t ❛s st✐♣✉❧❛t❡❞ ✐♥ ✷✱ ❛❢t❡r ♥♦t✐✜❝❛t✐♦♥ ❤❡ ♠✉st ✐♠♠❡❞✐❛t❡❧② ❧♦✇❡r t❤❡ ■♥t❡r♥❡t tr❛✣❝ t♦ t❤❡ ♥♦r♠❛❧ ❧❡✈❡❧✱ ❛♥❞ ♣❛② ❧❛t❡r t✇✐❝❡ ✭✷ ∗ [♣r✐❝❡]✮✳ ✹✳ ■❢ t❤❡ ❈❧✐❡♥t ❞♦❡s ♥♦t ❧♦✇❡r t❤❡ ■♥t❡r♥❡t tr❛✣❝ ✐♠♠❡❞✐❛t❡❧②✱ t❤❡♥ t❤❡ ❈❧✐❡♥t ✇✐❧❧ ❤❛✈❡ t♦ ♣❛② ✸ ∗ [♣r✐❝❡]✳ ✺✳ ❚❤❡ ❈❧✐❡♥t s❤❛❧❧✱ ❛s s♦♦♥ ❛s t❤❡ ■♥t❡r♥❡t ❙❡r✈✐❝❡ ❜❡❝♦♠❡s ♦♣❡r❛t✐✈❡✱ s✉❜♠✐t ✇✐t❤✐♥ s❡✈❡♥ ✭✼✮ ❞❛②s t❤❡ P❡rs♦♥❛❧ ❉❛t❛ ❋♦r♠ ❢r♦♠ ❤✐s ❛❝❝♦✉♥t ♦♥ t❤❡ Pr♦✈✐❞❡r✬s ✇❡❜ ♣❛❣❡ t♦ t❤❡ ❈❧✐❡♥t ❘❡❧❛t✐♦♥s ❉❡♣❛rt♠❡♥t ♦❢ t❤❡ Pr♦✈✐❞❡r✳
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷ ✴ ✹✷
✶ ❖❜t❛✐♥ ❛♥ ❡✲❝♦♥tr❛❝t ❢r♦♠ ❛ ❝♦♥✈❡♥t✐♦♥❛❧ ❝♦♥tr❛❝t
✷ ❲r✐t❡ t❤❡ ❡✲❝♦♥tr❛❝t ❞✐r❡❝t❧② ✐♥ ❛ ❢♦r♠❛❧ ❧❛♥❣✉❛❣❡
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸ ✴ ✹✷
✶ ❖❜t❛✐♥ ❛♥ ❡✲❝♦♥tr❛❝t ❢r♦♠ ❛ ❝♦♥✈❡♥t✐♦♥❛❧ ❝♦♥tr❛❝t
✷ ❲r✐t❡ t❤❡ ❡✲❝♦♥tr❛❝t ❞✐r❡❝t❧② ✐♥ ❛ ❢♦r♠❛❧ ❧❛♥❣✉❛❣❡
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸ ✴ ✹✷
✶
✷
✸
✹
✺
✻
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✹ ✴ ✹✷
✶ ●✐✈❡ ❛ ❢♦r♠❛❧ ❧❛♥❣✉❛❣❡ ❢♦r s♣❡❝✐❢②✐♥❣✴✇r✐t✐♥❣ ❝♦♥tr❛❝ts ✷ ❆♥❛❧②s❡ ❝♦♥tr❛❝ts ✏✐♥t❡r♥❛❧❧②✑
✸ ▼♦♥✐t♦r ❝♦♥tr❛❝ts
✹ ❉❡✈❡❧♦♣ ❛ t❤❡♦r② ♦❢ ❝♦♥tr❛❝ts
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✺ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✻ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✻ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✼ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✼ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✼ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✼ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✼ ✴ ✹✷
✶
✷
✸
✹
✺
✻
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✽ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✾ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✾ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✵ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✵ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✵ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✶ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✶ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✷ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✸ ✴ ✹✷
✶
✷
✸
✹
✺
✻
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✹ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✺ ✴ ✹✷
✶ P❝ ✐s s❡t ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ❝♦♥st❛♥ts ❖❛ ❛♥❞ F❛✱ ♦♥❡ ❢♦r ❡❛❝❤ ❜❛s✐❝
V ∩ ❖❛T V = ∅,
✷ ▼✉❧t✐s❡ts ♦❢ ❜❛s✐❝ ❛❝t✐♦♥s✿ ✐✳❡✳ γ = {❛, ❛, ❜} ✐s ❛ ❧❛❜❡❧ ✸ ❘❡str✐❝t❡❞ ♥♦♥✲❞❡t❡r♠✐♥✐s♠ ✭♠♦r❡ ❧❛t❡r✮
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✻ ✴ ✹✷
✶ P❝ ✐s s❡t ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ❝♦♥st❛♥ts ❖❛ ❛♥❞ F❛✱ ♦♥❡ ❢♦r ❡❛❝❤ ❜❛s✐❝
V ∩ ❖❛T V = ∅,
✷ ▼✉❧t✐s❡ts ♦❢ ❜❛s✐❝ ❛❝t✐♦♥s✿ ✐✳❡✳ γ = {❛, ❛, ❜} ✐s ❛ ❧❛❜❡❧ ✸ ❘❡str✐❝t❡❞ ♥♦♥✲❞❡t❡r♠✐♥✐s♠ ✭♠♦r❡ ❧❛t❡r✮
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✻ ✴ ✹✷
V = S
V = VPr♦♣(P)
V = V(❩) ❀ P❝T V = VPr♦♣(P❝)
V = S \ ϕT V
V = ϕT V ∩ ψT V
V = {s | ∀t ∈ S. (s, t) ∈ ❘γ ⇒ t ∈ ϕT V }
V = {❙ ⊆ S | ❙ ⊆ ϕT V[❩:=❙]}
V = ϕT V ∪ ψT V
V = {s | ∃t ∈ S. (s, t) ∈ ❘γ ∧ t ∈ ϕT V }
V = {❙ ⊆ S | ❙ ⊇ ϕT V[❩:=❙]}
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✼ ✴ ✹✷
✐=✶❛✐)) = {❛✶, . . . , ❛♥}(∧♥ ✐=✶❖❛✐)
✐=✶❛✐)) = {❛✶, . . . , ❛♥}(∧♥ ✐=✶¬F❛✐)
✐=✶❛✐)) = [{❛✶, . . . , ❛♥}](∧♥ ✐=✶F❛✐)
✐=✶❛✐]C) = [{❛✶, . . . , ❛♥}]❢ T (C)
✐=✶❛✐)α]C) = [{❛✶, . . . , ❛♥}]❢ T ([α]C)
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✽ ✴ ✹✷
✐=✶❛✐)) = {❛✶, . . . , ❛♥}(∧♥ ✐=✶❖❛✐)
✐=✶❛✐)) = [{❛✶, . . . , ❛♥}](∧♥ ✐=✶F❛✐)
✐=✶❛✐)) = {❛✶, . . . , ❛♥}(∧♥ ✐=✶¬F❛✐)
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✾ ✴ ✹✷
✐=✶❛✐)) = {❛✶, . . . , ❛♥}(∧♥ ✐=✶❖❛✐)
✐=✶❛✐)) = [{❛✶, . . . , ❛♥}](∧♥ ✐=✶F❛✐)
✐=✶❛✐)) = {❛✶, . . . , ❛♥}(∧♥ ✐=✶¬F❛✐)
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✾ ✴ ✹✷
✐=✶❛✐)) = {❛✶, . . . , ❛♥}(∧♥ ✐=✶❖❛✐)
✐=✶❛✐)) = [{❛✶, . . . , ❛♥}](∧♥ ✐=✶F❛✐)
✐=✶❛✐)) = {❛✶, . . . , ❛♥}(∧♥ ✐=✶¬F❛✐)
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✶✾ ✴ ✹✷
✶
✷
✸
✹
✺
✻
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✵ ✴ ✹✷
✶ ■t ✐s ♦❜❧✐❣❛t♦r② t❤❛t ♦♥❡ ♠❛✐❧s t❤❡ ❧❡tt❡r ✷ ■t ✐s ♦❜❧✐❣❛t♦r② t❤❛t ♦♥❡ ♠❛✐❧s t❤❡ ❧❡tt❡r ♦r ♦♥❡ ❞❡str♦②s t❤❡ ❧❡tt❡r
✶ ❖(♣) ✷ ❖(♣ ∨ q)
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✶ ✴ ✹✷
✶ ■t ✐s ♦❜❧✐❣❛t♦r② t❤❛t ♦♥❡ ♠❛✐❧s t❤❡ ❧❡tt❡r ✷ ■t ✐s ♦❜❧✐❣❛t♦r② t❤❛t ♦♥❡ ♠❛✐❧s t❤❡ ❧❡tt❡r ♦r ♦♥❡ ❞❡str♦②s t❤❡ ❧❡tt❡r
✶ ❖(♣) ✷ ❖(♣ ∨ q)
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✶ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✷ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✸ ✴ ✹✷
✶
✷
✸
✹
✺
✻
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✹ ✴ ✹✷
✶ ▼♦❞❡❧ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ❝♦♥tr❛❝t ✇r✐tt❡♥ ✐♥ ❊♥❣❧✐s❤ ✉s✐♥❣ t❤❡ ❢♦r♠❛❧
✷ ❚r❛♥s❧❛t❡ s②♥t❛❝t✐❝❛❧❧② t❤❡ CL s♣❡❝✐✜❝❛t✐♦♥ ✐♥t♦ t❤❡ ❡①t❡♥❞❡❞
✸ ❖❜t❛✐♥ ❛ ❑r✐♣❦❡✲❧✐❦❡ ♠♦❞❡❧ ✭❛ ❧❛❜❡❧❧❡❞ tr❛♥s✐t✐♦♥ s②st❡♠✱ ▲❚❙✮ ♦❢ t❤❡
✹ ❚r❛♥s❧❛t❡ t❤❡ ▲❚❙ ✐♥t♦ t❤❡ ✐♥♣✉t ❧❛♥❣✉❛❣❡ ♦❢ ◆✉❙▼❱❀ ✺ P❡r❢♦r♠ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ✉s✐♥❣ ◆✉❙▼❱❀ ✻ ■♥ ❝❛s❡ ♦❢ ❛ ❝♦✉♥t❡r✲❡①❛♠♣❧❡ ❣✐✈❡♥ ❜② ◆✉❙▼❱✱ ✐♥t❡r♣r❡t ✐t ❛s ❛ CL
✼ ❋✐♥❛❧❧②✱ r❡♣❛✐r t❤❡ ♦r✐❣✐♥❛❧ ❝♦♥tr❛❝t ❜② ❛❞❞✐♥❣ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧❛✉s❡✱
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✺ ✴ ✹✷
✶✳ ❚❤❡ ❈❧✐❡♥t s❤❛❧❧ ♥♦t s✉♣♣❧② ❢❛❧s❡ ✐♥❢♦r♠❛t✐♦♥ t♦ t❤❡ ❈❧✐❡♥t ❘❡❧❛t✐♦♥s ❉❡♣❛rt♠❡♥t ♦❢ t❤❡ Pr♦✈✐❞❡r✳ ✷✳ ❲❤❡♥❡✈❡r t❤❡ ■♥t❡r♥❡t ❚r❛✣❝ ✐s ❤✐❣❤ t❤❡♥ t❤❡ ❈❧✐❡♥t ♠✉st ♣❛② [♣r✐❝❡] ✐♠♠❡❞✐❛t❡❧②✱ ♦r t❤❡ ❈❧✐❡♥t ♠✉st ♥♦t✐❢② t❤❡ Pr♦✈✐❞❡r ❜② s❡♥❞✐♥❣ ❛♥ ❡✲♠❛✐❧ s♣❡❝✐❢②✐♥❣ t❤❛t ❤❡ ✇✐❧❧ ♣❛② ❧❛t❡r✳ ✸✳ ■❢ t❤❡ ❈❧✐❡♥t ❞❡❧❛②s t❤❡ ♣❛②♠❡♥t ❛s st✐♣✉❧❛t❡❞ ✐♥ ✷✱ ❛❢t❡r ♥♦t✐✜❝❛t✐♦♥ ❤❡ ♠✉st ✐♠♠❡❞✐❛t❡❧② ❧♦✇❡r t❤❡ ■♥t❡r♥❡t tr❛✣❝ t♦ t❤❡ ♥♦r♠❛❧ ❧❡✈❡❧✱ ❛♥❞ ♣❛② ❧❛t❡r t✇✐❝❡ ✭✷ ∗ [♣r✐❝❡]✮✳ ✹✳ ■❢ t❤❡ ❈❧✐❡♥t ❞♦❡s ♥♦t ❧♦✇❡r t❤❡ ■♥t❡r♥❡t tr❛✣❝ ✐♠♠❡❞✐❛t❡❧②✱ t❤❡♥ t❤❡ ❈❧✐❡♥t ✇✐❧❧ ❤❛✈❡ t♦ ♣❛② ✸ ∗ [♣r✐❝❡]✳ ✺✳ ❚❤❡ ❈❧✐❡♥t s❤❛❧❧✱ ❛s s♦♦♥ ❛s t❤❡ ■♥t❡r♥❡t ❙❡r✈✐❝❡ ❜❡❝♦♠❡s ♦♣❡r❛t✐✈❡✱ s✉❜♠✐t ✇✐t❤✐♥ s❡✈❡♥ ✭✼✮ ❞❛②s t❤❡ P❡rs♦♥❛❧ ❉❛t❛ ❋♦r♠ ❢r♦♠ ❤✐s ❛❝❝♦✉♥t ♦♥ t❤❡ Pr♦✈✐❞❡r✬s ✇❡❜ ♣❛❣❡ t♦ t❤❡ ❈❧✐❡♥t ❘❡❧❛t✐♦♥s ❉❡♣❛rt♠❡♥t ♦❢ t❤❡ Pr♦✈✐❞❡r✳ ✻✳ ❚❤❡ Pr♦✈✐❞❡r ♠❛②✱ ❛t ✐ts s♦❧❡ ❞✐s❝r❡t✐♦♥✱ ✇✐t❤♦✉t ♥♦t✐❝❡ ♦r ❣✐✈✐♥❣ ❛♥② r❡❛s♦♥ ♦r ✐♥❝✉rr✐♥❣ ❛♥② ❧✐❛❜✐❧✐t② ❢♦r ❞♦✐♥❣ s♦✿ ❙✉s♣❡♥❞ ■♥t❡r♥❡t ❙❡r✈✐❝❡s ✐♠♠❡❞✐❛t❡❧② ✐❢ ❈❧✐❡♥t ✐s ✐♥ ❜r❡❛❝❤ ♦❢ ❈❧❛✉s❡ ✶❀
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✻ ✴ ✹✷
✶ ❋P(s)(✜) ✷ [❤](φ ⇒ ❖(♣ + (❞&♥))) ✸ ([❞&♥](❖(❧) ∧ [❧]♦❖(♣&♣))) ✹ ([❞&♥ · ❧ ]♦❖(♣&♣&♣)) ✺ ([♦]❖(s❢❉))
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✼ ✴ ✹✷
✶ CL ✐♥t♦ Cµ
✶
✷
✸
✹
✺
✷ ❋r♦♠ Cµ t♦ ✐♥♣✉t ❧❛♥❣✉❛❣❡ ✐♥ ◆✉❙▼❱ ✭✉s✐♥❣ ❞✐r❡❝t s♣❡❝✐✜❝❛t✐♦♥✮ ✸ ▼♦❞❡❧ ❝❤❡❝❦✐♥❣ ✶
✷
✸
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✽ ✴ ✹✷
✶ CL ✐♥t♦ Cµ
✶
✷
✸
✹
✺
✷ ❋r♦♠ Cµ t♦ ✐♥♣✉t ❧❛♥❣✉❛❣❡ ✐♥ ◆✉❙▼❱ ✭✉s✐♥❣ ❞✐r❡❝t s♣❡❝✐✜❝❛t✐♦♥✮ ✸ ▼♦❞❡❧ ❝❤❡❝❦✐♥❣ ✶
✷
✸
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✽ ✴ ✹✷
✶ CL ✐♥t♦ Cµ
✶
✷
✸
✹
✺
✷ ❋r♦♠ Cµ t♦ ✐♥♣✉t ❧❛♥❣✉❛❣❡ ✐♥ ◆✉❙▼❱ ✭✉s✐♥❣ ❞✐r❡❝t s♣❡❝✐✜❝❛t✐♦♥✮ ✸ ▼♦❞❡❧ ❝❤❡❝❦✐♥❣ ✶
✷
✸
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✽ ✴ ✹✷
✶
✷
✸
✹
✺
✻
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✷✾ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✵ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✶ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✷ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✸ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✹ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✺ ✴ ✹✷
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✻ ✴ ✹✷
✶
✷
✸
✶
✷
✸
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✼ ✴ ✹✷
✶
✷
✸
✶
✷
✸
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✼ ✴ ✹✷
✶ ❨♦✉ ♠❛② ❡✐t❤❡r s❧❡❡♣ ♦♥ t❤❡ s♦❢❛ ♦r s❧❡❡♣ ♦♥ t❤❡ ❜❡❞✳ ✷ ❨♦✉ ♠❛② s❧❡❡♣ ♦♥ t❤❡ s♦❢❛ ❛♥❞ ②♦✉ ♠❛② s❧❡❡♣ ♦♥ t❤❡ ❜❡❞✳
✶ P(♣ ∨ q) ✷ P(♣) ∧ P(q)
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✽ ✴ ✹✷
✶ ■t ✐s ♦❜❧✐❣❛t♦r② ■ ♥♦✇ ♠❡❡t ❏♦♥❡s ✭❛s ♣r♦♠✐s❡❞ t♦ ❏♦♥❡s✮✳ ✷ ■t ✐s ♦❜❧✐❣❛t♦r② ■ ♥♦✇ ❞♦ ♥♦t ♠❡❡t ❏♦♥❡s ✭❛s ♣r♦♠✐s❡❞ t♦ ❙♠✐t❤✮✳
✶ ❖(♣) ✷ ❖(¬♣)
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✸✾ ✴ ✹✷
✶ ■t ♦✉❣❤t t♦ ❜❡ t❤❡ ❝❛s❡ t❤❛t ❏♦♥❡s ❤❡❧♣s ❙♠✐t❤ ✇❤♦ ❤❛s ❜❡❡♥ r♦❜❜❡❞✳ ✷ ■t ♦✉❣❤t t♦ ❜❡ t❤❡ ❝❛s❡ t❤❛t ❙♠✐t❤ ❤❛s ❜❡❡♥ r♦❜❜❡❞✳
✶ ❖(♣ ∧ q) ✷ ❖(q)
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✹✵ ✴ ✹✷
✶ ❏♦❤♥ ♦✉❣❤t t♦ ❣♦ t♦ t❤❡ ♣❛rt②✳ ✷ ■❢ ❏♦❤♥ ❣♦❡s t♦ t❤❡ ♣❛rt② t❤❡♥ ❤❡ ♦✉❣❤t t♦ t❡❧❧ t❤❡♠ ❤❡ ✐s ❝♦♠✐♥❣✳ ✸ ■❢ ❏♦❤♥ ❞♦❡s ♥♦t ❣♦ t♦ t❤❡ ♣❛rt② t❤❡♥ ❤❡ ♦✉❣❤t ♥♦t t♦ t❡❧❧ t❤❡♠ ❤❡ ✐s
✹ ❏♦❤♥ ❞♦❡s ♥♦t ❣♦ t♦ t❤❡ ♣❛rt②✳
✶ ❖(♣) ✷ ❖(♣ ⇒ q) ✸ ¬♣ ⇒ ❖(¬q) ✹ ¬♣
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✹✶ ✴ ✹✷
✶ ■t ✐s ♦❜❧✐❣❛t♦r② t❤❛t ❏♦❤♥ ❞♦❡s ♥♦t ❦✐❧❧ ❤✐s ♠♦t❤❡r✳ ✷ ■❢ ❏♦❤♥ ❞♦❡s ❦✐❧❧ ❤✐s ♠♦t❤❡r✱ t❤❡♥ ✐t ✐s ♦❜❧✐❣❛t♦r② t❤❛t ❏♦❤♥ ❦✐❧❧s ❤❡r
✸ ❏♦❤♥ ❞♦❡s ❦✐❧❧ ❤✐s ♠♦t❤❡r✳
✶ ❖(¬♣) ✷ ♣ ⇒ ❖(q) ✸ ♣
❆ ❋♦r♠❛❧ ▲❛♥❣✉❛❣❡ ❢♦r ❊✲❈♦♥tr❛❝ts ❊❞✐♥❜✉r❣❤✱ ✶✼✳✵✼✳✷✵✵✼ ✹✷ ✴ ✹✷