t r Ptr ts - - PowerPoint PPT Presentation
t r Ptr ts s tr s rs rt r rst
▼❛rt✐♥ ❙❝❤✇❛r✐❝❦✱ ❈❤r✐st✐❛♥ ❘♦❤r✱ ❋❡✐ ▲✐✉✱
❇r❛♥❞❡♥❜✉r❣ ❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② P❡tr✐ ◆❡ts ✷✵✷✵ ✲ P❛r✐s
✷✺ ❏✉♥❡ ✷✵✷✵
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶ ✴ ✸✼
✶
❲❤② ❝♦❧♦✉r❡❞ P❡tr✐ ◆❡ts❄
✷
❙t❛t❡ ♦❢ t❤❡ ❆rt
✸
❚❤❡ Pr♦❜❧❡♠
✹
❲❤❛t ❛r❡ ■❉❉❄
✺
■❉❉ ❜❛s✐❝ ♣r✐♥❝✐♣❧❡s
✻
■❉❉✲❜❛s❡❞ ✉♥❢♦❧❞✐♥❣ ❛❧❣♦r✐t❤♠
✼
❊①♣❡r✐♠❡♥ts
Unfolding
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸ ✴ ✸✼
❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿
zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿
zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿
zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿
zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
a=A & b=B : MassAction(k1)
colour-dependent rate function
❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿
zz ❙✐♠♣❧❡ ❈❙ ❡♥✉♠ ◆♦❞❡s ❂ ④❆✱❇✱❈✱❉✱❋✱●✱❍✱■✱❏ ⑥❀ zz Pr♦❞✉❝t ❈❙ ▼❛tr✐① ❂ Pr♦❞✭◆♦❞❡s✱◆♦❞❡s✮❀ zz ❙✉❜s❡t ❈❙ ❈♦♥♥❡❝t✐♦♥s ❂ ▼❛tr✐①❬✭❛❂❆ ✫ ✭❜❂❇⑤❜❂❈ ⑤❜❂❉ ⑤ ❜❂● ⑤ ❜❂■✮✮ ⑤ ✭❛❂❇ ✫ ✭❜❂❆⑤❜❂●⑤❜❂❍⑤❜❂❏✮✮ ⑤ ✭❛❂❈ ✫ ✭❜❂❋ ⑤ ❜❂❍✮✮❪❀ ✈❛r✐❛❜❧❡s✿ ◆♦❞❡s ✿ ❛❀ ◆♦❞❡s ✿ ❜❀ ❢✉♥❝t✐♦♥s✿ ❜♦♦❧ ■s❈♦♥♥❡❝t❡❞✭◆♦❞❡ ❛✶✱ ◆♦❞❡ ❜✶✮ ④✭❛✶✱❜✶✮ ❡❧❡♠❖❢ ❈♦♥♥❡❝t✐♦♥s⑥❀
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✹ ✴ ✸✼
Guards
❝♦❧♦✉r❡❞ ❝♦♥t✐♥✉♦✉s P❡tr✐ ♥❡t
Guard
❝♦❧♦✉r❡❞ st♦❝❤❛st✐❝ P❡tr✐ ♥❡t
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✺ ✴ ✸✼
❈♦❧♦✉r❡❞ P❡tr✐ ♥❡ts ❛r❡ ✐♥ ✉s❡ ❢♦r ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✱ ❝♦✈❡r✐♥❣ ♥❛t✉r❛❧✴❡♥❣✐♥❡❡r✐♥❣✴❧✐❢❡ s❝✐❡♥❝❡s✳ ❈✉rr❡♥t❧②✱ ♠♦st ❛♥❛❧②s✐s ❛♥❞ s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s r❡q✉✐r❡ ✉♥❢♦❧❞✐♥❣✿ ❝♦❧♦✉r❡❞ P❡tr✐ ♥❡t ♣❧❛✐♥ P❡tr✐ ♥❡t✳ ❯♥❢♦❧❞✐♥❣ t❡♥❞s t♦ ❜❡ t✐♠❡ ❝♦♥s✉♠✐♥❣✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✻ ✴ ✸✼
❈♦❧♦✉r❡❞ P❡tr✐ ♥❡ts ❛r❡ ✐♥ ✉s❡ ❢♦r ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✱ ❝♦✈❡r✐♥❣ ♥❛t✉r❛❧✴❡♥❣✐♥❡❡r✐♥❣✴❧✐❢❡ s❝✐❡♥❝❡s✳ ❈✉rr❡♥t❧②✱ ♠♦st ❛♥❛❧②s✐s ❛♥❞ s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s r❡q✉✐r❡ ✉♥❢♦❧❞✐♥❣✿ ❝♦❧♦✉r❡❞ P❡tr✐ ♥❡t Ñ ♣❧❛✐♥ P❡tr✐ ♥❡t✳ ❯♥❢♦❧❞✐♥❣ t❡♥❞s t♦ ❜❡ t✐♠❡ ❝♦♥s✉♠✐♥❣✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✻ ✴ ✸✼
❈♦❧♦✉r❡❞ P❡tr✐ ♥❡ts ❛r❡ ✐♥ ✉s❡ ❢♦r ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✱ ❝♦✈❡r✐♥❣ ♥❛t✉r❛❧✴❡♥❣✐♥❡❡r✐♥❣✴❧✐❢❡ s❝✐❡♥❝❡s✳ ❈✉rr❡♥t❧②✱ ♠♦st ❛♥❛❧②s✐s ❛♥❞ s✐♠✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s r❡q✉✐r❡ ✉♥❢♦❧❞✐♥❣✿ ❝♦❧♦✉r❡❞ P❡tr✐ ♥❡t Ñ ♣❧❛✐♥ P❡tr✐ ♥❡t✳ ❯♥❢♦❧❞✐♥❣ t❡♥❞s t♦ ❜❡ t✐♠❡ ❝♦♥s✉♠✐♥❣✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✻ ✴ ✸✼
❊①❛♠♣❧❡ ♦❢ ❛ s❝❛❧❡❛❜❧❡ ♠♦❞❡❧ t♦ ❛❞❥✉st ❣r✐❞ s✐③❡✳ ✷❉ ❉✐✛✉s✐♦♥ ✐♥ s♣❛❝❡✳
pos 10 10`(MID,MID) Grid2D diffusion [IsNeighbour2D4(x,y,a,b)] (x,y) (a,b)
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✼ ✴ ✸✼
❊①❛♠♣❧❡ ♦❢ ❛ s❝❛❧❡❛❜❧❡ ♠♦❞❡❧ t♦ ❛❞❥✉st ❣r✐❞ s✐③❡✳ ✷❉ ❉✐✛✉s✐♦♥ ✐♥ s♣❛❝❡✳
pos 10 10`(MID,MID) Grid2D diffusion [IsNeighbour2D4(x,y,a,b)] (x,y) (a,b)
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✼ ✴ ✸✼
❚❤❡ ❝♦r❡ ♣r♦❜❧❡♠ ♦❢ ❡✣❝✐❡♥t ✉♥❢♦❧❞✐♥❣ ✐s t♦ ❞❡t❡r♠✐♥❡ t❤❡ tr❛♥s✐t✐♦♥ ✐♥st❛♥❝❡s✱ ❡✳❣✱ ❛❧❧ ❜✐♥❞✐♥❣s ♦❢ t❤❡ ✐♥✈♦❧✈❡❞ ✈❛r✐❛❜❧❡s✳
pos 10 10`(MID,MID) Grid2D diffusion [IsNeighbour2D4(x,y,a,b)] (x,y) (a,b)
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✽ ✴ ✸✼
❚❤❡ ❝♦r❡ ♣r♦❜❧❡♠ ♦❢ ❡✣❝✐❡♥t ✉♥❢♦❧❞✐♥❣ ✐s t♦ ❞❡t❡r♠✐♥❡ t❤❡ tr❛♥s✐t✐♦♥ ✐♥st❛♥❝❡s✱ ❡✳❣✱ ❛❧❧ ❜✐♥❞✐♥❣s ♦❢ t❤❡ ✐♥✈♦❧✈❡❞ ✈❛r✐❛❜❧❡s✳
pos 10 10`(MID,MID) Grid2D diffusion [IsNeighbour2D4(x,y,a,b)] (x,y) (a,b)
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✾ ✴ ✸✼
❚❤❡ ❝♦r❡ ♣r♦❜❧❡♠ ♦❢ ❡✣❝✐❡♥t ✉♥❢♦❧❞✐♥❣ ✐s t♦ ❞❡t❡r♠✐♥❡ t❤❡ tr❛♥s✐t✐♦♥ ✐♥st❛♥❝❡s✱ ❡✳❣✱ ❛❧❧ ❜✐♥❞✐♥❣s ♦❢ t❤❡ ✐♥✈♦❧✈❡❞ ✈❛r✐❛❜❧❡s✳ Range of Variables x, y, a and b 1 .. D
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✵ ✴ ✸✼
■♥t❡r✈❛❧ ❞❡❝✐s✐♦♥ ❞✐❛❣r❛♠s Ñ ❈❙P✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✶ ✴ ✸✼
❉✐r❡❝t❡❞ ❛❝②❝❧✐❝ ❣r❛♣❤s ✭❉❆●s✮ t♦ ❡♥❝♦❞❡ ✐♥t❡r✈❛❧ ❧♦❣✐❝ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ❢♦r♠ ♦❢ s②♠❜♦❧✐❝ ❞❛t❛ str✉❝t✉r❡✳
1 x1 [0,6) x2 [6,8) [0,1) [1,oo) [8,oo)
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✷ ✴ ✸✼
❚❤❡② ❤❛✈❡ t✇♦ t②♣❡s ♦❢ ♥♦❞❡s✿ ♥♦♥✲t❡r♠✐♥❛❧ ✭❡❧❧✐♣s❡s✮ ❛♥❞ t❡r♠✐♥❛❧ ♦♥❡s ✭❜♦①❡s✮✳
x1 x2
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✷ ✴ ✸✼
❚❤❡② ❤❛✈❡ t✇♦ t②♣❡s ♦❢ ♥♦❞❡s✿ ♥♦♥✲t❡r♠✐♥❛❧ ✭❡❧❧✐♣s❡s✮ ❛♥❞ t❡r♠✐♥❛❧ ♦♥❡s ✭❜♦①❡s✮✳
1 x1 x2
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✷ ✴ ✸✼
◆♦♥✲t❡r♠✐♥❛❧ ♥♦❞❡s ♠❛② ❤❛✈❡ ❛♥ ❛r❜✐tr❛r② ♥✉♠❜❡r ♦❢ ♦✉t❣♦✐♥❣ ❛r❝s ❧❛❜❡❧❧❡❞ ✇✐t❤ ✐♥t❡r✈❛❧s ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ✐♥ t❤❡ ❢♦r♠ ❬❛✱❜✮✳
1 x1 [0,6) x2 [6,8) [0,1) [1,oo) [8,oo)
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✷ ✴ ✸✼
❚❤❡ s❡t ♦❢ ❛❧❧ ♣❛t❤s ❣♦✐♥❣ ❢r♦♠✿ t❤❡ r♦♦t Ñ t❤❡ t❡r♠✐♥❛❧ ♥♦❞❡ ✶ ❞❡s❝r✐❜❡s ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❣✐✈❡♥ ❝♦♥str❛✐♥t ♣r♦❜❧❡♠✳ ❚②♣✐❝❛❧❧②✱ ♦♥❡ ♣❛t❤ ❡♥❝♦❞❡s ♠♦r❡ t❤❛♥ ♦♥❡ s♦❧✉t✐♦♥✳ ❚❤✉s✱ ✇❡ ❝❛♥ ❡❛s✐❧② ♣✐❝❦ ❛❧❧ ❈❙P s♦❧✉t✐♦♥s ❢r♦♠ t❤❡ ❝♦♥str❛✐♥t ■❉❉✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✸ ✴ ✸✼
❚❤❡ s❡t ♦❢ ❛❧❧ ♣❛t❤s ❣♦✐♥❣ ❢r♦♠✿ t❤❡ r♦♦t Ñ t❤❡ t❡r♠✐♥❛❧ ♥♦❞❡ ✶ ❞❡s❝r✐❜❡s ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❣✐✈❡♥ ❝♦♥str❛✐♥t ♣r♦❜❧❡♠✳ ❚②♣✐❝❛❧❧②✱ ♦♥❡ ♣❛t❤ ❡♥❝♦❞❡s ♠♦r❡ t❤❛♥ ♦♥❡ s♦❧✉t✐♦♥✳ ❚❤✉s✱ ✇❡ ❝❛♥ ❡❛s✐❧② ♣✐❝❦ ❛❧❧ ❈❙P s♦❧✉t✐♦♥s ❢r♦♠ t❤❡ ❝♦♥str❛✐♥t ■❉❉✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✸ ✴ ✸✼
❚❤❡ s❡t ♦❢ ❛❧❧ ♣❛t❤s ❣♦✐♥❣ ❢r♦♠✿ t❤❡ r♦♦t Ñ t❤❡ t❡r♠✐♥❛❧ ♥♦❞❡ ✶ ❞❡s❝r✐❜❡s ❛❧❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❣✐✈❡♥ ❝♦♥str❛✐♥t ♣r♦❜❧❡♠✳ ❚②♣✐❝❛❧❧②✱ ♦♥❡ ♣❛t❤ ❡♥❝♦❞❡s ♠♦r❡ t❤❛♥ ♦♥❡ s♦❧✉t✐♦♥✳ ❚❤✉s✱ ✇❡ ❝❛♥ ❡❛s✐❧② ♣✐❝❦ ❛❧❧ ❈❙P s♦❧✉t✐♦♥s ❢r♦♠ t❤❡ ❝♦♥str❛✐♥t ■❉❉✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✸ ✴ ✸✼
❱❛r✐❛❜❧❡ ♦r❞❡r✐♥❣
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✹ ✴ ✸✼
①✶ ě ✽✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✹ ✴ ✸✼
s♦♠❡ ✐♥t❡r♠❡❞✐❛t❡ s❝r❡❡♥s❤♦ts✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✹ ✴ ✸✼
❖♥❡ ✜♥❛❧ s♦❧✉t✐♦♥✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✹ ✴ ✸✼
■♥t❡r✈❛❧ ♣❛rt✐t✐♦♥s ❧❛❜❡❧❧✐♥❣ t❤❡ ♦✉t❣♦✐♥❣ ❛r❝s ♦❢ ❡❛❝❤ ♥♦♥✲t❡r♠✐♥❛❧ ♥♦❞❡ ❛r❡ r❡❞✉❝❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ r✻, ✼q ❛♥❞ r✼, ✽s ❊❛❝❤ ♥♦♥✲t❡r♠✐♥❛❧ ♥♦❞❡ ❤❛s ❛t ❧❡❛st t✇♦ ❞✐✛❡r❡♥t ❝❤✐❧❞r❡♥✳ ❚❤❡r❡ ❡①✐st ♥♦ t✇♦ ♥♦❞❡s ✇✐t❤ ✐s♦♠♦r♣❤✐❝ s✉❜❣r❛♣❤s✳
♥♦t ❘❡❞✉❝❡❞ ❘❡❞✉❝❡❞
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✺ ✴ ✸✼
■♥t❡r✈❛❧ ♣❛rt✐t✐♦♥s ❧❛❜❡❧❧✐♥❣ t❤❡ ♦✉t❣♦✐♥❣ ❛r❝s ♦❢ ❡❛❝❤ ♥♦♥✲t❡r♠✐♥❛❧ ♥♦❞❡ ❛r❡ r❡❞✉❝❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ r✻, ✼q ❛♥❞ r✼, ✽s ❊❛❝❤ ♥♦♥✲t❡r♠✐♥❛❧ ♥♦❞❡ ❤❛s ❛t ❧❡❛st t✇♦ ❞✐✛❡r❡♥t ❝❤✐❧❞r❡♥✳ ❚❤❡r❡ ❡①✐st ♥♦ t✇♦ ♥♦❞❡s ✇✐t❤ ✐s♦♠♦r♣❤✐❝ s✉❜❣r❛♣❤s✳
♥♦t ❘❡❞✉❝❡❞ ❘❡❞✉❝❡❞
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✺ ✴ ✸✼
■♥t❡r✈❛❧ ♣❛rt✐t✐♦♥s ❧❛❜❡❧❧✐♥❣ t❤❡ ♦✉t❣♦✐♥❣ ❛r❝s ♦❢ ❡❛❝❤ ♥♦♥✲t❡r♠✐♥❛❧ ♥♦❞❡ ❛r❡ r❡❞✉❝❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ r✻, ✼q ❛♥❞ r✼, ✽s ❊❛❝❤ ♥♦♥✲t❡r♠✐♥❛❧ ♥♦❞❡ ❤❛s ❛t ❧❡❛st t✇♦ ❞✐✛❡r❡♥t ❝❤✐❧❞r❡♥✳ ❚❤❡r❡ ❡①✐st ♥♦ t✇♦ ♥♦❞❡s ✇✐t❤ ✐s♦♠♦r♣❤✐❝ s✉❜❣r❛♣❤s✳
1 x1 [0,6) x2 [7,8) x2 [6,7) [0,1) [1,oo) [0,1) [1,oo) [8,oo)
♥♦t ❘❡❞✉❝❡❞
1 x1 [0,6) x2 [6,8) [0,1) [1,oo) [8,oo)
❘❡❞✉❝❡❞
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✺ ✴ ✸✼
Pr❡♣❛r❛t✐♦♥ st❡♣✿ r❡❣✐str❛t✐♦♥ ♦❢ ❝♦♥st❛♥ts✱ ❝♦❧♦r s❡ts✱ ✈❛r✐❛❜❧❡s ❛♥❞ ❢✉♥❝t✐♦♥s✳ ❯♥❢♦❧❞ ♣❧❛❝❡s ❛♥❞ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✿ t❤✐s ♠❛② ✐♥✈♦❧✈❡ ❈❙P ✐❢ ✇❡ ❤❛✈❡ s✉❜s❡t ♠❡❝❤❛♥✐s♠ ❛♥❞ t♦ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✳ ❯♥❢♦❧❞ tr❛♥s✐t✐♦♥s✿ ❛♥❞ t❤❡✐r ❛❞❥❛❝❡♥t ❛r❝s ❛♥❞ ❛❞❞ t❤❡ r❡s✉❧t t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❆❞❞ ❛❧❧ ✉♥❢♦❧❞❡❞ ♣❧❛❝❡s ✇❤✐❝❤ ❛r❡ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ tr❛♥s✐t✐♦♥ ✉♥❢♦❧❞✐♥❣s t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❚❤✐s ✐♠♣❧✐❝✐t❧② r❡♠♦✈❡s ✐s♦❧❛t❡❞ ♣❧❛❝❡s✳ ❚❤❡ ❡♥t✐r❡ ♣s❡✉❞♦ ❝♦❞❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✐s ❣✐✈❡♥ ✐♥ t❤❡ ♣❛♣❡r✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✻ ✴ ✸✼
Pr❡♣❛r❛t✐♦♥ st❡♣✿ r❡❣✐str❛t✐♦♥ ♦❢ ❝♦♥st❛♥ts✱ ❝♦❧♦r s❡ts✱ ✈❛r✐❛❜❧❡s ❛♥❞ ❢✉♥❝t✐♦♥s✳ ❯♥❢♦❧❞ ♣❧❛❝❡s ❛♥❞ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✿ t❤✐s ♠❛② ✐♥✈♦❧✈❡ ❈❙P ✐❢ ✇❡ ❤❛✈❡ s✉❜s❡t ♠❡❝❤❛♥✐s♠ ❛♥❞ t♦ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✳ ❯♥❢♦❧❞ tr❛♥s✐t✐♦♥s✿ ❛♥❞ t❤❡✐r ❛❞❥❛❝❡♥t ❛r❝s ❛♥❞ ❛❞❞ t❤❡ r❡s✉❧t t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❆❞❞ ❛❧❧ ✉♥❢♦❧❞❡❞ ♣❧❛❝❡s ✇❤✐❝❤ ❛r❡ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ tr❛♥s✐t✐♦♥ ✉♥❢♦❧❞✐♥❣s t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❚❤✐s ✐♠♣❧✐❝✐t❧② r❡♠♦✈❡s ✐s♦❧❛t❡❞ ♣❧❛❝❡s✳ ❚❤❡ ❡♥t✐r❡ ♣s❡✉❞♦ ❝♦❞❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✐s ❣✐✈❡♥ ✐♥ t❤❡ ♣❛♣❡r✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✻ ✴ ✸✼
Pr❡♣❛r❛t✐♦♥ st❡♣✿ r❡❣✐str❛t✐♦♥ ♦❢ ❝♦♥st❛♥ts✱ ❝♦❧♦r s❡ts✱ ✈❛r✐❛❜❧❡s ❛♥❞ ❢✉♥❝t✐♦♥s✳ ❯♥❢♦❧❞ ♣❧❛❝❡s ❛♥❞ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✿ t❤✐s ♠❛② ✐♥✈♦❧✈❡ ❈❙P ✐❢ ✇❡ ❤❛✈❡ s✉❜s❡t ♠❡❝❤❛♥✐s♠ ❛♥❞ t♦ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✳ ❯♥❢♦❧❞ tr❛♥s✐t✐♦♥s✿ ❛♥❞ t❤❡✐r ❛❞❥❛❝❡♥t ❛r❝s ❛♥❞ ❛❞❞ t❤❡ r❡s✉❧t t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❆❞❞ ❛❧❧ ✉♥❢♦❧❞❡❞ ♣❧❛❝❡s ✇❤✐❝❤ ❛r❡ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ tr❛♥s✐t✐♦♥ ✉♥❢♦❧❞✐♥❣s t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❚❤✐s ✐♠♣❧✐❝✐t❧② r❡♠♦✈❡s ✐s♦❧❛t❡❞ ♣❧❛❝❡s✳ ❚❤❡ ❡♥t✐r❡ ♣s❡✉❞♦ ❝♦❞❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✐s ❣✐✈❡♥ ✐♥ t❤❡ ♣❛♣❡r✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✻ ✴ ✸✼
Pr❡♣❛r❛t✐♦♥ st❡♣✿ r❡❣✐str❛t✐♦♥ ♦❢ ❝♦♥st❛♥ts✱ ❝♦❧♦r s❡ts✱ ✈❛r✐❛❜❧❡s ❛♥❞ ❢✉♥❝t✐♦♥s✳ ❯♥❢♦❧❞ ♣❧❛❝❡s ❛♥❞ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✿ t❤✐s ♠❛② ✐♥✈♦❧✈❡ ❈❙P ✐❢ ✇❡ ❤❛✈❡ s✉❜s❡t ♠❡❝❤❛♥✐s♠ ❛♥❞ t♦ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✳ ❯♥❢♦❧❞ tr❛♥s✐t✐♦♥s✿ ❛♥❞ t❤❡✐r ❛❞❥❛❝❡♥t ❛r❝s ❛♥❞ ❛❞❞ t❤❡ r❡s✉❧t t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❆❞❞ ❛❧❧ ✉♥❢♦❧❞❡❞ ♣❧❛❝❡s ✇❤✐❝❤ ❛r❡ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ tr❛♥s✐t✐♦♥ ✉♥❢♦❧❞✐♥❣s t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❚❤✐s ✐♠♣❧✐❝✐t❧② r❡♠♦✈❡s ✐s♦❧❛t❡❞ ♣❧❛❝❡s✳ ❚❤❡ ❡♥t✐r❡ ♣s❡✉❞♦ ❝♦❞❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✐s ❣✐✈❡♥ ✐♥ t❤❡ ♣❛♣❡r✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✻ ✴ ✸✼
Pr❡♣❛r❛t✐♦♥ st❡♣✿ r❡❣✐str❛t✐♦♥ ♦❢ ❝♦♥st❛♥ts✱ ❝♦❧♦r s❡ts✱ ✈❛r✐❛❜❧❡s ❛♥❞ ❢✉♥❝t✐♦♥s✳ ❯♥❢♦❧❞ ♣❧❛❝❡s ❛♥❞ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✿ t❤✐s ♠❛② ✐♥✈♦❧✈❡ ❈❙P ✐❢ ✇❡ ❤❛✈❡ s✉❜s❡t ♠❡❝❤❛♥✐s♠ ❛♥❞ t♦ ❞❡t❡r♠✐♥❡ ✐♥✐t✐❛❧ ♠❛r❦✐♥❣✳ ❯♥❢♦❧❞ tr❛♥s✐t✐♦♥s✿ ❛♥❞ t❤❡✐r ❛❞❥❛❝❡♥t ❛r❝s ❛♥❞ ❛❞❞ t❤❡ r❡s✉❧t t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❆❞❞ ❛❧❧ ✉♥❢♦❧❞❡❞ ♣❧❛❝❡s ✇❤✐❝❤ ❛r❡ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ tr❛♥s✐t✐♦♥ ✉♥❢♦❧❞✐♥❣s t♦ t❤❡ ✉♥❢♦❧❞❡❞ ♥❡t✳ ❚❤✐s ✐♠♣❧✐❝✐t❧② r❡♠♦✈❡s ✐s♦❧❛t❡❞ ♣❧❛❝❡s✳ ❚❤❡ ❡♥t✐r❡ ♣s❡✉❞♦ ❝♦❞❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✐s ❣✐✈❡♥ ✐♥ t❤❡ ♣❛♣❡r✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✻ ✴ ✸✼
p1 p2 p3 t
❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿
❝s ❂ ④✶✱✽✱✸✳✳✻✱✶✵✱✾✱✶✶✱✷✵✳✳✷✸⑥❀ ❡♥✉♠ ❛❜ ❂ ④❆✱❈✱❉⑥❀ ✈❛r✐❛❜❧❡s✿ ❝s ✿ ①❀ ❛❜ ✿ ②❀
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✼ ✴ ✸✼
p1 13 1`all() cs p2 cs p3 1`all() ab t x x y
❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿
❝s ❂ ④✶✱✽✱✸✳✳✻✱✶✵✱✾✱✶✶✱✷✵✳✳✷✸⑥❀ ❡♥✉♠ ❛❜ ❂ ④❆✱❈✱❉⑥❀ ✈❛r✐❛❜❧❡s✿ ❝s ✿ ①❀ ❛❜ ✿ ②❀
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✽ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✽ ✴ ✸✼
p1 13 1`all() cs p2 cs p3 1`all() ab t [(6<=x&x<=10)&(y=A)] x x y
❈♦❧♦r ❞❡✜♥✐t✐♦♥s✿
❝s ❂ ④✶✱✽✱✸✳✳✻✱✶✵✱✾✱✶✶✱✷✵✳✳✷✸⑥❀ ❡♥✉♠ ❛❜ ❂ ④❆✱❈✱❉⑥❀ ✈❛r✐❛❜❧❡s✿ ❝s ✿ ①❀ ❛❜ ✿ ②❀
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✶✾ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✵ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✶ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✷ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✸ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✹ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✺ ✴ ✸✼
❚✇♦✲♣❛t❤ s♦❧✉t✐♦♥✿
✶st ♣❛t❤ ✿ ✭② ❂ ❆✱① ❂ ✻✮ ✷♥❞ ♣❛t❤ ✿ ✭② ❂ ❆✱ ① ❂ ✽ ✳✳✶✵✮
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✻ ✴ ✸✼
p1_6 p1_8 p1_9 p1_10 p2_6 p2_8 p2_9 p2_10 p3_A t_0_6 t_0_8 t_0_9 t_0_10 ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✼ ✴ ✸✼
p1_6 p1_8 p1_9 p1_10 p2_6 p2_8 p2_9 p2_10 p3_A t_0_6 t_0_8 t_0_9 t_0_10 ❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✼ ✴ ✸✼
❲❡ ❝♦♠♣❛r❡❞ ♦✉r ■❉❉ ✉♥❢♦❧❞✐♥❣ ✇✐t❤ ❛♥ ✉♥❢♦❧❞✐♥❣ ❡♠♣❧♦②✐♥❣ t❤❡ ♣♦♣✉❧❛r ❝♦♥str❛✐♥t s♦❧✈❡r ❧✐❜r❛r② ●❡❝♦❞❡✳ ✷✷ ▼❈❈ ♠♦❞❡❧s ✭P◆▼▲ ❢♦r♠❛t✮ ❤tt♣s✿✴✴♠❝❝✳❧✐♣✻✳❢r✴♠♦❞❡❧s✳♣❤♣
✶st ❣r♦✉♣✿ r❡q✉✐r❡s ♥♦ s✉❜st❛♥t✐❛❧ ✉♥❢♦❧❞✐♥❣ t✐♠❡✳ ✷♥❞ ❣r♦✉♣✿ r❡q✉✐r❡s s✉❜st❛♥t✐❛❧ ✉♥❢♦❧❞✐♥❣ t✐♠❡✳
❲❡ ✉s❡❞ ❛❧s♦ t✇♦ ❜✐♦❧♦❣✐❝❛❧ t❡st ❝❛s❡s ❢r♦♠ ♦✉r ♦✇♥ ❝♦❧❧❡❝t✐♦♥✿
❤tt♣s✿✴✴✇✇✇✲❞ss③✳✐♥❢♦r♠❛t✐❦✳t✉✲❝♦tt❜✉s✳❞❡✴❉❙❙❩✴❙♦❢t✇❛r❡✴❊①❛♠♣❧❡s❄❞✐r❂■❞❞❯♥❢♦❧❞✐♥❣
✸❉ ❉✐✛✉s✐♦♥✳ ❇r✉ss❡❧❛t♦r✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✽ ✴ ✸✼
❲❡ ❝♦♠♣❛r❡❞ ♦✉r ■❉❉ ✉♥❢♦❧❞✐♥❣ ✇✐t❤ ❛♥ ✉♥❢♦❧❞✐♥❣ ❡♠♣❧♦②✐♥❣ t❤❡ ♣♦♣✉❧❛r ❝♦♥str❛✐♥t s♦❧✈❡r ❧✐❜r❛r② ●❡❝♦❞❡✳ ✷✷ ▼❈❈ ♠♦❞❡❧s ✭P◆▼▲ ❢♦r♠❛t✮ Ñ ❤tt♣s✿✴✴♠❝❝✳❧✐♣✻✳❢r✴♠♦❞❡❧s✳♣❤♣
✶st ❣r♦✉♣✿ r❡q✉✐r❡s ♥♦ s✉❜st❛♥t✐❛❧ ✉♥❢♦❧❞✐♥❣ t✐♠❡✳ ✷♥❞ ❣r♦✉♣✿ r❡q✉✐r❡s s✉❜st❛♥t✐❛❧ ✉♥❢♦❧❞✐♥❣ t✐♠❡✳
❲❡ ✉s❡❞ ❛❧s♦ t✇♦ ❜✐♦❧♦❣✐❝❛❧ t❡st ❝❛s❡s ❢r♦♠ ♦✉r ♦✇♥ ❝♦❧❧❡❝t✐♦♥✿
❤tt♣s✿✴✴✇✇✇✲❞ss③✳✐♥❢♦r♠❛t✐❦✳t✉✲❝♦tt❜✉s✳❞❡✴❉❙❙❩✴❙♦❢t✇❛r❡✴❊①❛♠♣❧❡s❄❞✐r❂■❞❞❯♥❢♦❧❞✐♥❣
✸❉ ❉✐✛✉s✐♦♥✳ ❇r✉ss❡❧❛t♦r✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✽ ✴ ✸✼
❲❡ ❝♦♠♣❛r❡❞ ♦✉r ■❉❉ ✉♥❢♦❧❞✐♥❣ ✇✐t❤ ❛♥ ✉♥❢♦❧❞✐♥❣ ❡♠♣❧♦②✐♥❣ t❤❡ ♣♦♣✉❧❛r ❝♦♥str❛✐♥t s♦❧✈❡r ❧✐❜r❛r② ●❡❝♦❞❡✳ ✷✷ ▼❈❈ ♠♦❞❡❧s ✭P◆▼▲ ❢♦r♠❛t✮ Ñ ❤tt♣s✿✴✴♠❝❝✳❧✐♣✻✳❢r✴♠♦❞❡❧s✳♣❤♣
✶st ❣r♦✉♣✿ r❡q✉✐r❡s ♥♦ s✉❜st❛♥t✐❛❧ ✉♥❢♦❧❞✐♥❣ t✐♠❡✳ ✷♥❞ ❣r♦✉♣✿ r❡q✉✐r❡s s✉❜st❛♥t✐❛❧ ✉♥❢♦❧❞✐♥❣ t✐♠❡✳
❲❡ ✉s❡❞ ❛❧s♦ t✇♦ ❜✐♦❧♦❣✐❝❛❧ t❡st ❝❛s❡s ❢r♦♠ ♦✉r ♦✇♥ ❝♦❧❧❡❝t✐♦♥✿
❤tt♣s✿✴✴✇✇✇✲❞ss③✳✐♥❢♦r♠❛t✐❦✳t✉✲❝♦tt❜✉s✳❞❡✴❉❙❙❩✴❙♦❢t✇❛r❡✴❊①❛♠♣❧❡s❄❞✐r❂■❞❞❯♥❢♦❧❞✐♥❣
✸❉ ❉✐✛✉s✐♦♥✳ ❇r✉ss❡❧❛t♦r✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✽ ✴ ✸✼
t❤❡ ♠♦❞❡❧
❛ ❧❛♥❡ ❜r✐❞❣❡ ✇✐t❤ ❧✐♠✐t❡❞ ❝❛♣❛❝✐t②✳ ✉s❡❞ ❜② t✇♦ t②♣❡s ♦❢ ✈❡❤✐❝❧❡s✳ ❝♦❧♦✉r❡❞ ♠♦❞❡❧ ❤❛s ✶✺ ♣❧❛❝❡s✱ ✶✶ tr❛♥s✐t✐♦♥ ❛♥❞ ✺✼ ❛r❝s✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✾ ✴ ✸✼
2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 N 0.0 0.5 1.0 1.5 2.0 2.5 time(sec) IDD Gecode
N | P | | T | | A | ✶ ✷✽ ✺✷ ✸✷✻ ✷ ✹✽ ✷✽✽ ✷✵✾✵ ✹ ✼✽ ✾✻✽ ✼✸✺✵ ✺ ✶✵✽ ✷✷✷✽ ✶✼ ✶✾✵ ✾ ✶✷✽ ✶✸✷✽ ✶✵ ✵✶✵ ✶✵ ✶✸✽ ✷✸✹✽ ✶✽ ✵✾✵ ✶✶ ✶✻✽ ✺✹✵✽ ✹✷ ✸✸✵ ✶✺ ✶✽✽ ✷✶✵✽ ✶✺ ✾✺✵ ✶✻ ✶✾✽ ✸✼✷✽ ✷✽ ✽✸✵ ✷✵ ✷✷✽ ✽✺✽✽ ✻✼ ✹✼✵ ❋✐❣✉r❡✿ ❇r✐❞❣❡s ❛♥❞ ✈❡❤✐❝❧❡s ✭▼❈❈✮❀ r❡q✉✐r❡s ♥♦ s✉❜st❛♥t✐❛❧ ✉♥❢♦❧❞✐♥❣ t✐♠❡
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✷✾ ✴ ✸✼
t❤❡ ♠♦❞❡❧
r❡✉♥✐✜❝❛t✐♦♥ ♣r♦❝❡ss✳ t❤❡ ❝♦❧♦✉r❡❞ ♠♦❞❡❧ ❤❛s ✶✵✹ ♣❧❛❝❡s✱ ✻✻ tr❛♥s✐t✐♦♥ ❛♥❞✶✾✽ ❛r❝s✳ ✐t ✐s s❝❛❧❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ❧❡❣❛❧ r❡s✐❞❡♥ts✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✵ ✴ ✸✼
200 400 600 800 1000 1200 N 50 100 150 200 250 300 time(sec) IDD Gecode
N | P | | T | | A | ✶✵ ✶✹✼✺ ✶✷✸✹ ✸✼✾✾ ✷✵ ✸✷✼✶ ✷✼✺✸ ✽✹✹✻ ✺✵ ✶✷ ✶✾✹ ✶✵ ✺✻✵ ✸✷ ✷✸✽ ✶✵✵ ✹✵ ✻✵✺ ✸✻ ✽✼✶ ✶✶✷ ✼✷✽ ✷✵✵ ✶✹✸ ✾✵✽ ✶✸✹ ✷✼✾ ✹✶✶ ✹✻✾ ✹✵✵ ✺✸✼ ✼✵✽ ✺✵✽ ✹✽✾ ✶ ✺✺✽ ✼✷✾ ✽✵✵ ✷ ✵✼✺ ✸✵✽ ✶ ✾✼✻ ✾✵✾ ✻ ✵✻✶ ✷✹✾ ✶✷✵✵ ✹ ✻✶✷ ✾✵✽ ✹ ✹✵✺ ✸✷✾ ✶✸ ✺✵✼ ✼✻✾ ❋✐❣✉r❡✿ ❋❛♠✐❧② ❘❡✉♥✐♦♥ ✭▼❈❈✮❀ r❡q✉✐r❡s s✉❜st❛♥t✐❛❧ ✉♥❢♦❧❞✐♥❣ t✐♠❡
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✵ ✴ ✸✼
pos 10 10`(MID,MID,MID) Grid3D diffusion [IsNeighbour(x,y,z,a,b,c)] (x,y,z) (a,b,c)
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✶ ✴ ✸✼
10 20 30 40 50 N 20 40 60 80 time(sec) IDD Gecode
N | P | | T | | A | ✺ ✶✷✺ ✻✵✵ ✶✷✵✵ ✶✵ ✶✵✵✵ ✺✹✵✵ ✶✵ ✽✵✵ ✶✺ ✸✸✼✺ ✶✽ ✾✵✵ ✸✼ ✽✵✵ ✷✵ ✽✵✵✵ ✼✷ ✾✾✽ ✶✹✺ ✾✾✻ ✷✺ ✶✺ ✻✷✺ ✾✵ ✵✵✵ ✶✽✵ ✵✵✵ ✸✵ ✷✼ ✵✵✵ ✶✺✻ ✻✵✵ ✸✶✸ ✷✵✵ ✸✺ ✹✷ ✽✼✺ ✷✹✾ ✾✵✵ ✹✾✾ ✽✵✵ ✹✵ ✻✹ ✵✵✵ ✸✼✹ ✹✵✵ ✼✹✽ ✽✵✵ ✹✺ ✾✶ ✶✷✺ ✺✸✹ ✻✵✵ ✶ ✵✻✾ ✷✵✵ ✺✵ ✶✷✺ ✵✵✵ ✼✸✺ ✵✵✵ ✶ ✹✼✵ ✵✵✵ ❋✐❣✉r❡✿ ❉✐✛✉s✐♦♥ ✭✸❉✮❀ ◆ ✕ ●r✐❞ s✐③❡
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✶ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✷ ✴ ✸✼
50 100 150 200 250 N 100 200 300 400 time(sec) IDD Gecode
N | P | | T | | A | ✷✺ ✶✷✺✵ ✶✶ ✾✵✽ ✷✸ ✶✾✶ ✺✵ ✺✵✵✵ ✹✽ ✽✵✽ ✾✺ ✶✶✻ ✼✺ ✶✶ ✷✺✵ ✶✶✵ ✼✵✽ ✷✶✺ ✼✾✶ ✶✵✵ ✷✵ ✵✵✵ ✶✾✼ ✻✵✽ ✸✽✺ ✷✶✻ ✶✷✺ ✸✶ ✷✺✵ ✸✵✾ ✺✵✽ ✻✵✸ ✸✾✶ ✶✺✵ ✹✺ ✵✵✵ ✹✹✻ ✹✵✽ ✽✼✵ ✸✶✻ ✶✼✺ ✻✶ ✷✺✵ ✻✵✽ ✸✵✽ ✶ ✶✽✺ ✾✾✶ ✷✵✵ ✽✵ ✵✵✵ ✼✾✺ ✷✵✽ ✶ ✺✺✵ ✹✶✻ ✷✷✺ ✶✵✶ ✷✺✵ ✶ ✵✵✼ ✶✵✽ ✶ ✾✻✸ ✺✾✶ ✷✺✵ ✶✷✺ ✵✵✵ ✶ ✷✹✹ ✵✵✽ ✷ ✹✷✺ ✺✶✻ ❋✐❣✉r❡✿ ❇r✉ss❡❧❛t♦r ❀ ◆ ✕ ●r✐❞ s✐③❡ ♦❢ ❛ ✷❉ sq✉❛r❡
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✷ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✸ ✴ ✸✼
♠♦❞❡❧s ✇✐t❤ ❛ ❢❡✇ ❣✉❛r❞s ♦r ♥♦ ❣✉❛r❞s✳ ♠♦❞❡❧s ✇✐t❤ s✐♠♣❧❡ ❝♦❧♦✉r s❡ts✳ ❡✳❣✱ ✶✷ ▼❈❈ ♠♦❞❡❧s ✭♥♦ s✉❜st❛♥t✐❛❧ t✐♠❡✮✳
■❉❉s✱ ✇❤❡♥✿
♣❛r❛♠❡tr✐③❡❞ ♠♦❞❡❧s ✇✐t❤ ❛ ❧❛r❣❡ s❝❛❧✐♥❣ ❢❛❝t♦r✱ ❡✳❣✱ ❉✐✛✉s✐♦♥ ✐♥ s♣❛❝❡✳ ♠♦❞❡❧s ✇✐t❤ s♦♣❤✐st✐❝❛t❡❞ ❣✉❛r❞s✳ ❡✳❣✱ ♠♦st ♠♦❞❡❧s ✐♥ ♦✉r ♦✇♥ ❝♦❧❧❡❝t✐♦♥✳
❚❤❡ ❝♦♠♣❧❡t❡ ♣❡r❢♦r♠❛♥❝❡ r❡♣♦rt ✐s ❛✈❛✐❧❛❜❧❡✿
❤tt♣s✿✴✴✇✇✇✲❞ss③✳✐♥❢♦r♠❛t✐❦✳t✉✲ ❝♦tt❜✉s✳❞❡✴❉❙❙❩✴❙♦❢t✇❛r❡✴❊①❛♠♣❧❡s❄❞✐r❂■❞❞❯♥❢♦❧❞✐♥❣
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✹ ✴ ✸✼
♠♦❞❡❧s ✇✐t❤ ❛ ❢❡✇ ❣✉❛r❞s ♦r ♥♦ ❣✉❛r❞s✳ ♠♦❞❡❧s ✇✐t❤ s✐♠♣❧❡ ❝♦❧♦✉r s❡ts✳ ❡✳❣✱ ✶✷ ▼❈❈ ♠♦❞❡❧s ✭♥♦ s✉❜st❛♥t✐❛❧ t✐♠❡✮✳
■❉❉s✱ ✇❤❡♥✿
♣❛r❛♠❡tr✐③❡❞ ♠♦❞❡❧s ✇✐t❤ ❛ ❧❛r❣❡ s❝❛❧✐♥❣ ❢❛❝t♦r✱ ❡✳❣✱ ❉✐✛✉s✐♦♥ ✐♥ s♣❛❝❡✳ ♠♦❞❡❧s ✇✐t❤ s♦♣❤✐st✐❝❛t❡❞ ❣✉❛r❞s✳ ❡✳❣✱ ♠♦st ♠♦❞❡❧s ✐♥ ♦✉r ♦✇♥ ❝♦❧❧❡❝t✐♦♥✳
❚❤❡ ❝♦♠♣❧❡t❡ ♣❡r❢♦r♠❛♥❝❡ r❡♣♦rt ✐s ❛✈❛✐❧❛❜❧❡✿
❤tt♣s✿✴✴✇✇✇✲❞ss③✳✐♥❢♦r♠❛t✐❦✳t✉✲ ❝♦tt❜✉s✳❞❡✴❉❙❙❩✴❙♦❢t✇❛r❡✴❊①❛♠♣❧❡s❄❞✐r❂■❞❞❯♥❢♦❧❞✐♥❣
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✹ ✴ ✸✼
♠♦❞❡❧s ✇✐t❤ ❛ ❢❡✇ ❣✉❛r❞s ♦r ♥♦ ❣✉❛r❞s✳ ♠♦❞❡❧s ✇✐t❤ s✐♠♣❧❡ ❝♦❧♦✉r s❡ts✳ ❡✳❣✱ ✶✷ ▼❈❈ ♠♦❞❡❧s ✭♥♦ s✉❜st❛♥t✐❛❧ t✐♠❡✮✳
■❉❉s✱ ✇❤❡♥✿
♣❛r❛♠❡tr✐③❡❞ ♠♦❞❡❧s ✇✐t❤ ❛ ❧❛r❣❡ s❝❛❧✐♥❣ ❢❛❝t♦r✱ ❡✳❣✱ ❉✐✛✉s✐♦♥ ✐♥ s♣❛❝❡✳ ♠♦❞❡❧s ✇✐t❤ s♦♣❤✐st✐❝❛t❡❞ ❣✉❛r❞s✳ ❡✳❣✱ ♠♦st ♠♦❞❡❧s ✐♥ ♦✉r ♦✇♥ ❝♦❧❧❡❝t✐♦♥✳
❚❤❡ ❝♦♠♣❧❡t❡ ♣❡r❢♦r♠❛♥❝❡ r❡♣♦rt ✐s ❛✈❛✐❧❛❜❧❡✿
❤tt♣s✿✴✴✇✇✇✲❞ss③✳✐♥❢♦r♠❛t✐❦✳t✉✲ ❝♦tt❜✉s✳❞❡✴❉❙❙❩✴❙♦❢t✇❛r❡✴❊①❛♠♣❧❡s❄❞✐r❂■❞❞❯♥❢♦❧❞✐♥❣
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✹ ✴ ✸✼
❤tt♣s✿✴✴✇✇✇✲❞ss③✳✐♥❢♦r♠❛t✐❦✳t✉✲❝♦tt❜✉s✳❞❡✴❉❙❙❩✴❙♦❢t✇❛r❡✴
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✺ ✴ ✸✼
CANDL CANDL
❤tt♣s✿✴✴✇✇✇✲❞ss③✳✐♥❢♦r♠❛t✐❦✳t✉✲❝♦tt❜✉s✳❞❡✴❉❙❙❩✴❙♦❢t✇❛r❡✴
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✺ ✴ ✸✼
❤tt♣s✿✴✴✇✇✇✲❞ss③✳✐♥❢♦r♠❛t✐❦✳t✉✲❝♦tt❜✉s✳❞❡✴❉❙❙❩✴❙♦❢t✇❛r❡✴
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✺ ✴ ✸✼
P❡r❢♦r♠❛♥❝❡✿
❈♦♥s✐❞❡r✐♥❣ ♠❡♠♦r②✳ P♦✇❡r ❝♦♥s✉♠♣t✐♦♥✳
■♠♣❧❡♠❡♥t❛t✐♦♥ ❡✣❝✐❡♥❝②✿
▼✉❧t✐✲t❤r❡❛❞✐♥❣✿ ✉♥❢♦❧❞✐♥❣ t❤❡ ❝♦❧♦✉r❡❞ ♣❧❛❝❡s ❛♥❞ tr❛♥s✐t✐♦♥ ✐s ❝✉rr❡♥t❧② ❞♦♥❡ s❡q✉❡♥t✐❛❧❧②✳ ❘❡✉s❡ ♦❢ ❛❧r❡❛❞② ❝♦♠♣✉t❡❞ s♦❧✉t✐♦♥s✳ ❈❤♦♦s✐♥❣ ❛♠♦♥❣ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡ ♦r❞❡r str❛t❡❣✐❡s✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✻ ✴ ✸✼
P❡r❢♦r♠❛♥❝❡✿
❈♦♥s✐❞❡r✐♥❣ ♠❡♠♦r②✳ P♦✇❡r ❝♦♥s✉♠♣t✐♦♥✳
■♠♣❧❡♠❡♥t❛t✐♦♥ ❡✣❝✐❡♥❝②✿
▼✉❧t✐✲t❤r❡❛❞✐♥❣✿ ✉♥❢♦❧❞✐♥❣ t❤❡ ❝♦❧♦✉r❡❞ ♣❧❛❝❡s ❛♥❞ tr❛♥s✐t✐♦♥ ✐s ❝✉rr❡♥t❧② ❞♦♥❡ s❡q✉❡♥t✐❛❧❧②✳ ❘❡✉s❡ ♦❢ ❛❧r❡❛❞② ❝♦♠♣✉t❡❞ s♦❧✉t✐♦♥s✳ ❈❤♦♦s✐♥❣ ❛♠♦♥❣ s❡✈❡r❛❧ ✈❛r✐❛❜❧❡ ♦r❞❡r str❛t❡❣✐❡s✳
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✻ ✴ ✸✼
❙❝❤✇❛r✐❝❦✱ ❘♦❤r✱ ▲✐✉✱ ❆ss❛❢✱ ❈❤♦❞❛❦✱ ❍❡✐♥❡r ✷✺ ❏✉♥❡ ✷✵✷✵ ✸✼ ✴ ✸✼