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slide-1
SLIDE 1

▲♦❣✐❝ ❛♥❞ str✉❝t✉r❡s ❢♦r s②st❡♠ ♠♦❞❡❧❧✐♥❣

❚❡♦❞♦r ❑♥❛♣✐❦

❯♥✐✈❡rs✐té ❞❡ ❧❛ ◆♦✉✈❡❧❧❡ ❈❛❧❡❞♦♥✐❡ ✫ ■▼❙❝

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶ ✴ ✾✼

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SLIDE 2

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-3
SLIDE 3

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-4
SLIDE 4

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-5
SLIDE 5

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-6
SLIDE 6

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-7
SLIDE 7

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-8
SLIDE 8

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-9
SLIDE 9

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-10
SLIDE 10

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-11
SLIDE 11

■♥tr♦❞✉❝t✐♦♥

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♠♦❞❡❧ ❛ s②st❡♠ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ✐ts ❜❡❤❛✈✐♦✉r

❡✳❣✳ ❛tt❡st t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s s♦♠❡t❤✐♥❣ ✇r♦♥❣ ✇✐❧❧ ♥❡✈❡r ❤❛♣♣❡♥

✇❤❡r❡ ✏s②st❡♠✑ ♠❛② ❜❡ ❡✳❣✳ ❞✐❣✐t❛❧ ❝✐r❝✉✐t ❝♦♠♣✉t❡r s②st❡♠✿ ❤❛r❞✇❛r❡✱ s♦❢t✇❛r❡✱ ♥❡t✇♦r❦ ♥❛t✉r❛❧ s②st❡♠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷ ✴ ✾✼

slide-12
SLIDE 12

Pr❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ s②st❡♠

✶ ❛ss❡rt ❛ ♣r♦♣❡rt② ♦❢ ❛ s②st❡♠ ✷ ♣r♦✈❡ t❤❛t t❤❡ ❛ss❡rt✐♦♥ ❤♦❧❞s ❢♦r t❤❡ s②st❡♠

◗ ✿ ❆r❡ t❤❡r❡ ❛❧❣♦r✐t❤♠s ❢♦r ✭✷✮ ❄ ❆ ✿ ■t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❧❛ss ♦❢ s②st❡♠s✱ t❤❡ ❝❧❛ss ♦❢ ♣r♦♣❡rt✐❡s✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸ ✴ ✾✼

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SLIDE 13

Pr❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ s②st❡♠

✶ ❛ss❡rt ❛ ♣r♦♣❡rt② ♦❢ ❛ s②st❡♠ ✷ ♣r♦✈❡ t❤❛t t❤❡ ❛ss❡rt✐♦♥ ❤♦❧❞s ❢♦r t❤❡ s②st❡♠

◗ ✿ ❆r❡ t❤❡r❡ ❛❧❣♦r✐t❤♠s ❢♦r ✭✷✮ ❄ ❆ ✿ ■t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❧❛ss ♦❢ s②st❡♠s✱ t❤❡ ❝❧❛ss ♦❢ ♣r♦♣❡rt✐❡s✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸ ✴ ✾✼

slide-14
SLIDE 14

Pr❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ s②st❡♠

✶ ❛ss❡rt ❛ ♣r♦♣❡rt② ♦❢ ❛ s②st❡♠ ✷ ♣r♦✈❡ t❤❛t t❤❡ ❛ss❡rt✐♦♥ ❤♦❧❞s ❢♦r t❤❡ s②st❡♠

◗ ✿ ❆r❡ t❤❡r❡ ❛❧❣♦r✐t❤♠s ❢♦r ✭✷✮ ❄ ❆ ✿ ■t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❧❛ss ♦❢ s②st❡♠s✱ t❤❡ ❝❧❛ss ♦❢ ♣r♦♣❡rt✐❡s✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸ ✴ ✾✼

slide-15
SLIDE 15

Pr❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ s②st❡♠

✶ ❛ss❡rt ❛ ♣r♦♣❡rt② ♦❢ ❛ s②st❡♠ ✷ ♣r♦✈❡ t❤❛t t❤❡ ❛ss❡rt✐♦♥ ❤♦❧❞s ❢♦r t❤❡ s②st❡♠

◗ ✿ ❆r❡ t❤❡r❡ ❛❧❣♦r✐t❤♠s ❢♦r ✭✷✮ ❄ ❆ ✿ ■t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❧❛ss ♦❢ s②st❡♠s✱ t❤❡ ❝❧❛ss ♦❢ ♣r♦♣❡rt✐❡s✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸ ✴ ✾✼

slide-16
SLIDE 16

Pr❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ s②st❡♠

✶ ❛ss❡rt ❛ ♣r♦♣❡rt② ♦❢ ❛ s②st❡♠ ✷ ♣r♦✈❡ t❤❛t t❤❡ ❛ss❡rt✐♦♥ ❤♦❧❞s ❢♦r t❤❡ s②st❡♠

◗ ✿ ❆r❡ t❤❡r❡ ❛❧❣♦r✐t❤♠s ❢♦r ✭✷✮ ❄ ❆ ✿ ■t ❞❡♣❡♥❞s ♦♥

  • t❤❡ ❝❧❛ss ♦❢ s②st❡♠s✱
  • t❤❡ ❝❧❛ss ♦❢ ♣r♦♣❡rt✐❡s✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸ ✴ ✾✼

slide-17
SLIDE 17

Pr❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ s②st❡♠

✶ ❛ss❡rt ❛ ♣r♦♣❡rt② ♦❢ ❛ s②st❡♠ ✷ ♣r♦✈❡ t❤❛t t❤❡ ❛ss❡rt✐♦♥ ❤♦❧❞s ❢♦r t❤❡ s②st❡♠

◗ ✿ ❆r❡ t❤❡r❡ ❛❧❣♦r✐t❤♠s ❢♦r ✭✷✮ ❄ ❆ ✿ ■t ❞❡♣❡♥❞s ♦♥

  • t❤❡ ❝❧❛ss ♦❢ s②st❡♠s✱
  • t❤❡ ❝❧❛ss ♦❢ ♣r♦♣❡rt✐❡s✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸ ✴ ✾✼

slide-18
SLIDE 18

❚✇♦ ✇❛②s ♦❢ s②st❡♠ ♠♦❞❡❧❧✐♥❣ ✭❛♠♦♥❣ ♦t❤❡rs✮

✶ ❆①✐♦♠❛t✐❝ ♠❡t❤♦❞

s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ s❡t ♦❢ s❡♥t❡♥❝❡s ✱ ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ✱ ❄ ✭ ❡✳❣✳ r❡s♦❧✉t✐♦♥✮

✷ ▼♦❞❡❧✲t❤❡♦r❡t✐❝ ♠❡t❤♦❞

s②st❡♠ ✐s ❣✐✈❡♥ ❛s str✉❝t✉r❡ ✱ ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ✱ ❄

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹ ✴ ✾✼

slide-19
SLIDE 19

❚✇♦ ✇❛②s ♦❢ s②st❡♠ ♠♦❞❡❧❧✐♥❣ ✭❛♠♦♥❣ ♦t❤❡rs✮

✶ ❆①✐♦♠❛t✐❝ ♠❡t❤♦❞

  • s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ s❡t ♦❢ s❡♥t❡♥❝❡s Φ✱
  • ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ✱
  • Φ ⊢ ϕ ❄

✭⇔ Φ ∪ {¬ϕ} ⊢ ◻ ❡✳❣✳ r❡s♦❧✉t✐♦♥✮

✷ ▼♦❞❡❧✲t❤❡♦r❡t✐❝ ♠❡t❤♦❞

s②st❡♠ ✐s ❣✐✈❡♥ ❛s str✉❝t✉r❡ ✱ ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ✱ ❄

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹ ✴ ✾✼

slide-20
SLIDE 20

❚✇♦ ✇❛②s ♦❢ s②st❡♠ ♠♦❞❡❧❧✐♥❣ ✭❛♠♦♥❣ ♦t❤❡rs✮

✶ ❆①✐♦♠❛t✐❝ ♠❡t❤♦❞

  • s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ s❡t ♦❢ s❡♥t❡♥❝❡s Φ✱
  • ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ✱
  • Φ ⊢ ϕ ❄

✭⇔ Φ ∪ {¬ϕ} ⊢ ◻ ❡✳❣✳ r❡s♦❧✉t✐♦♥✮

✷ ▼♦❞❡❧✲t❤❡♦r❡t✐❝ ♠❡t❤♦❞

  • s②st❡♠ ✐s ❣✐✈❡♥ ❛s str✉❝t✉r❡ A✱
  • ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ✱
  • A ⊧ ϕ ❄

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹ ✴ ✾✼

slide-21
SLIDE 21

❚✇♦ ✇❛②s ♦❢ s②st❡♠ ♠♦❞❡❧❧✐♥❣ ✭❛♠♦♥❣ ♦t❤❡rs✮

✶ ❆①✐♦♠❛t✐❝ ♠❡t❤♦❞

  • s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ s❡t ♦❢ s❡♥t❡♥❝❡s Φ✱
  • ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ✱
  • Φ ⊢ ϕ ❄

✭⇔ Φ ∪ {¬ϕ} ⊢ ◻ ❡✳❣✳ r❡s♦❧✉t✐♦♥✮

✷ ▼♦❞❡❧✲t❤❡♦r❡t✐❝ ♠❡t❤♦❞

  • s②st❡♠ ✐s ❣✐✈❡♥ ❛s str✉❝t✉r❡ A✱
  • ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ✱
  • A ⊧ ϕ ❄

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹ ✴ ✾✼

slide-22
SLIDE 22

❊①❛♠♣❧❡ ✭❛ ♥❛✐✈❡ ♦♥❡✮

❍❡r❜✐✈♦rs

4 3 5 2 6 1 ⋯

birth death birth death birth death death birth death birth

Pr❡❞❛t♦rs

4 3 5 2 6 1 ⋯

eating eating eating eating eating birth death birth death birth death death birth death birth

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺ ✴ ✾✼

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SLIDE 23

❊①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮

■♥t❡r❛❝t✐♦♥ Pr❡❞❛t♦rs ❍❡r❜✐✈♦rs ❡❛t✐♥❣ ❞❡❛t❤ (e,d) ❜✐rt❤ (b,=) ❞❡❛t❤ (d,=) ❜✐rt❤ (=,b) ❞❡❛t❤ (=,d)

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻ ✴ ✾✼

slide-24
SLIDE 24

❊①❛♠♣❧❡ ✭s②♥❝❤r♦♥✐s❡❞ ♣r♦❞✉❝t✮

(1, 1) (2, 1) (3, 1) (4, 1) (b, =) (d, =) (b, =) (d, =) (b, =) (d, =) (1, 2) (2, 2) (3, 2) (4, 2) (=, b) (=, d) (e, d) (=, b) (=, d) (e, d) (=, b) (=, d) (e, d) (=, b) (=, d) (e, d) (b, =) (d, =) (b, =) (d, =) (b, =) (d, =) (1, 3) (2, 3) (3, 3) (4, 3) (=, b) (=, d) (e, d) (=, b) (=, d) (e, d) (=, b) (=, d) (e, d) (=, b) (=, d) (e, d) (b, =) (d, =) (b, =) (d, =) (b, =) (d, =) ⋮ ⋮ ⋮ ⋮ ⋯ ⋯ ⋯

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼ ✴ ✾✼

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SLIDE 25

❙tr✉❝t✉r❡s

❆❧♣❤❛❜❡t Σ ✇✐t❤ ❛r✐t✐❡s α∶Σ → IN ❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ✭✶✮ A∶Σ → ⋃n∈α(Σ) ℘(An) ❝♦✉♥t❛❜❧❡ ♦r ✜♥✐t❡ ✉♥✐✈❡rs❡ A ❢♦r ❡❛❝❤ a ∈ Σ✱ A(a) ✐s ❛ r❡❧❛t✐♦♥ ♦❢ ❛r✐t② α(a) A(a) ⊆ Aα(a) ❙♣❡❝✐❛❧ ❝❛s❡s max(α(Σ)) ≤ 2

↝ A ✐s ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ ✇❤❡r❡ ❜✐♥❛r② r❡❧❛t✐♦♥ s②♠❜♦❧s ❛r❡ ❡❞❣❡ ❧❛❜❡❧s ❛♥❞ ✉♥❛r② r❡❧❛t✐♦♥ s②♠❜♦❧s ❛r❡ ✈❡rt❡① ❧❛❜❡❧s

α(a) > 2

↝ A(a) ♠❛② ❜❡ s❡❡♥ ❛s ❛ s❡t ♦❢ a✕❧❛❜❡❧❧❡❞ ❤②♣❡r❡❞❣❡s✱ ↝ A ✐s ❛ ✭❞✐r❡❝t❡❞✮ ❤②♣❡r❣r❛♣❤

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽ ✴ ✾✼

slide-26
SLIDE 26

❊①❛♠♣❧❡s

✶ (e,d),(b,−),(d,−),(−,b)(−,d) ❛r❡ ❜✐♥❛r② r❡❧❛t✐♦♥s

↝ ❞✐r❡❝t❡❞ ❣r❛♣❤

✷ t❡r♥❛r② r❡❧❛t✐♦♥ ✇✐t❤ t✉♣❧❡s ✭❢❛t❤❡r✱ ♠♦t❤❡r✱ ❝❤✐❧❞✮

↝ ❡✈❡r② t✉♣❧❡ ✐s ❛ ❤②♣❡r❡❞❣❡ ↝ ❞✐r❡❝t❡❞ ❤②♣❡r❣r❛♣❤

Ben Am y Dev Cha y a Esha F arukh Harish Ja y Gina Kani Ila

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾ ✴ ✾✼

slide-27
SLIDE 27

❲♦r❦✐♥❣ ❡①❛♠♣❧❡

Σ = {likes,talks} α(likes) = 2✱ α(talks) = 3 A = {Amy,Ben,Chaya,Dora,Elil,Farukh}

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✵ ✴ ✾✼

slide-28
SLIDE 28

A(❧✐❦❡s) ❛s ❛ t❛❜❧❡

❧✐❦❡s

❧✐❦✐♥❣ ❧✐❦❡❞

❆♠② ❆♠② ❆♠② ❇❡♥ ❆♠② ❈❤❛②❛ ❆♠② ❉♦r❛ ❆♠② ❋❛r✉❦❤ ❇❡♥ ❆♠② ❇❡♥ ❈❤❛②❛ ❇❡♥ ❉♦r❛ ❇❡♥ ❊❧✐❧ ❇❡♥ ❋❛r✉❦❤ ❈❤❛②❛ ❆♠② ❈❤❛②❛ ❇❡♥ ❈❤❛②❛ ❈❤❛②❛ ❈❤❛②❛ ❉♦r❛ ❈❤❛②❛ ❊❧✐❧ ❈❤❛②❛ ❋❛r✉❦❤ ❉♦r❛ ❈❤❛②❛ ❉♦r❛ ❉♦r❛ ❉♦r❛ ❋❛r✉❦❤ ❊❧✐❧ ❆♠② ❊❧✐❧ ❈❤❛②❛ ❊❧✐❧ ❊❧✐❧ ❋❛r✉❦❤ ❆♠② ❋❛r✉❦❤ ❇❡♥ ❋❛r✉❦❤ ❈❤❛②❛ ❋❛r✉❦❤ ❉♦r❛ ❋❛r✉❦❤ ❊❧✐❧ ❋❛r✉❦❤ ❋❛r✉❦❤

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✶ ✴ ✾✼

slide-29
SLIDE 29

❋✐rst✲♦r❞❡r q✉❡r✐❡s✿ ❧♦♦❦✐♥❣ ❢♦r s♦❧✉t✐♦♥s ♦❢ ❢♦r♠✉❧❛❡

likes(x,x)

likes(x,y) ∧ x = y

∀y likes(x,y)

∃x likes(x,y)

¬∃y likes(x,y)

✻ likes(x,y) ∧ likes(y,x) ∧ ¬likes(x,x) ✼ ∀y (∀z likes(y,z) ⇒ likes(x,y)) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✷ ✴ ✾✼

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SLIDE 30

[ ∣likes(x,x)∣ ] = {❆♠②✱ ❈❤❛②❛✱ ❉♦r❛✱ ❊❧✐❧✱ ❋❛r✉❦❤} = ❆♠② ❈❤❛②❛ ❉♦r❛ ❊❧✐❧ ❋❛r✉❦❤

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✸ ✴ ✾✼

slide-31
SLIDE 31

[ ∣likes(x,y) ∧ x = y∣ ] = { ✭❆♠②✱ ❆♠②✮✱ ✭❈❤❛②❛✱ ❈❤❛②❛✮✱ ✭❉♦r❛✱ ❉♦r❛✮✱ ✭❊❧✐❧✱ ❊❧✐❧✮✱ ✭❋❛r✉❦❤✱ ❋❛r✉❦❤✮ ⑥ = ❆♠② ❆♠② ❈❤❛②❛ ❈❤❛②❛ ❉♦r❛ ❉♦r❛ ❊❧✐❧ ❊❧✐❧ ❋❛r✉❦❤ ❋❛r✉❦❤

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✹ ✴ ✾✼

slide-32
SLIDE 32

[ ∣∀y likes(x,y)∣ ] = {❈❤❛②❛✱ ❋❛r✉❦❤} = ❈❤❛②❛ ❋❛r✉❦❤

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✺ ✴ ✾✼

slide-33
SLIDE 33

[ ∣∃x likes(x,y)∣ ] = ❆♠② ❇❡♥ ❈❤❛②❛ ❉♦r❛ ❊❧✐❧ ❋❛r✉❦❤

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✻ ✴ ✾✼

slide-34
SLIDE 34

✺ [

∣¬∃y likes(x,y)∣ ] = ∅

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✼ ✴ ✾✼

slide-35
SLIDE 35

[ ∣likes(x,y) ∧ likes(y,x) ∧ ¬likes(x,x)∣ ] = { ✭❇❡♥✱ ❆♠②✮✱ ✭❇❡♥✱ ❈❤❛②❛✮✱ ✭❇❡♥✱ ❋❛r✉❦❤✮ ⑥ = ❇❡♥ ❆♠② ❇❡♥ ❈❤❛②❛ ❇❡♥ ❋❛r✉❦❤

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✽ ✴ ✾✼

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SLIDE 36

[ ∣∀y (∀z likes(y,z) ⇒ likes(x,y))∣ ] = { ❆♠②✱ ❇❡♥✱ ❈❤❛②❛✱ ❉♦r❛✱ ❋❛r✉❦❤ ⑥ = ❆♠② ❇❡♥ ❈❤❛②❛ ❉♦r❛ ❋❛r✉❦❤

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✾ ✴ ✾✼

slide-37
SLIDE 37

❊①♣❧✐❝✐t ✭❤②♣❡r✮❡❞❣❡s✿ ✐♥❝✐❞❡♥❝❡ str✉❝t✉r❡s

❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ✭✶✮ ✶ ✉♥✐✈❡rs❡ ↝ ✐♠♣❧✐❝✐t ✭❤②♣❡r✮❡❞❣❡s ❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ✭✷✮ ✷ ✉♥✐✈❡rs❡s ✿ A✈❡rt✱ A❡❞❣ A = ⟨A✈❡rt,A❧❛❜⟩ A✈❡rt∶A❡❞❣ → ⋃n∈α(Σ) An

✈❡rt

A❧❛❜∶A❡❞❣ → Σ s✳t✳ A✈❡rt(e) ∈ Aα(A❧❛❜(e))

✈❡rt

❢♦r ❡✈❡r② e ∈ A❡❞❣

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✵ ✴ ✾✼

slide-38
SLIDE 38

❘❡♠❛r❦ ❛❜♦✉t ✶st ♦r❞❡r ❛t♦♠✐❝ ❢♦r♠✉❧❛s

❋♦r ❡❛❝❤ a ∈ Σ ✇✐t❤ α(a) = n t②♣❡ ✶ str✉❝t✉r❡s a(x1,...,xn) a(x) t②♣❡ ✷ str✉❝t✉r❡s ❡❞❣a(y,x1,...,xn) y r❛♥❣❡s ♦✈❡r A❡❞❣ x1,...,xn r❛♥❣❡ ♦✈❡r A✈❡rt

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✶ ✴ ✾✼

slide-39
SLIDE 39

❊①♣r❡ss✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ❛ str✉❝t✉r❡

❆ ♣r♦♣❡rt② P ✐s ❡①♣r❡ss✐❜❧❡ ✐♥ ❛ ❧♦❣✐❝ L ✐❢ t❤❡r❡ ❡①✐sts ❛♥ L✲s❡♥t❡♥❝❡ ϕ s✳t✳✱ ❢♦r ❡✈❡r② str✉❝t✉r❡ A A ❤❛s ♣r♦♣❡rt② P ⇔ A ⊧ ϕ ❆ r❡❧❛t✐♦♥ r ⊆ An ✐s ❞❡✜♥❛❜❧❡ ✐♥ L ✐❢ t❤❡r❡ ❡①✐sts ❛♥ L✲❢♦r♠✉❧❛ ϕ(x) ✇✐t❤ ❢r❡❡ ✈❛r✐❛❜❧❡s x = (x1,...,xn) s✳t✳ r = {d ∈ An ∣ A ⊧ ϕ(d)}

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✷ ✴ ✾✼

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SLIDE 40

❋❖ ✿ ✜rst ♦r❞❡r ❧♦❣✐❝

❊①❛♠♣❧❡s ♦❢ ♣r♦♣❡rt✐❡s ❢♦r t②♣❡ ✶ str✉❝t✉r❡s ❋✐♥✐t❡ ❛✉t♦♠❛t♦♥ ♦✈❡r {a,b,c} ✐s ♥♦r♠❛❧✐③❡❞ ✿ Σ = {a,b,c,ι,f} ✇✐t❤ α(a) = α(b) = α(c) = 2✱ α(ι) = α(f) = 1

∃x (ι(x) ∧ ∀x′ (ι(x′) ⇒ x′ = x) ∧ ∀z ¬(a(z,x) ∨ b(z,x) ∨ c(z,x))) ∧ ∃y (f(y) ∧ ∀y′ (f(y′) ⇒ y′ = y) ∧ ∀z ¬(a(y,z) ∨ b(y,z) ∨ c(y,z)))

❊✈❡r② r❡❧❛t✐♦♥ ✐s ✏❢✉♥❝t✐♦♥♥❛❧✑ ✇✳r✳t✳ t♦ ✐ts ❧❛st ❝♦♠♣♦♥❡♥t ✿ Σ = {f,g,h} ✇✐t❤ α(f) = 2✱ α(g) = α(h) = 3

∀x ∀y1 ∀y2 ((f(x,y1) ∧ f(x,y2)) ⇒ y1 = y2) ∧ ∀x ∀y ∀z1 ∀z2 ((g(x,y,z1) ∧ g(x,y,z2)) ⇒ z1 = z2) ∧ ∀x ∀y ∀z1 ∀z2 ((h(x,y,z1) ∧ h(x,y,z2)) ⇒ z1 = z2)

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✸ ✴ ✾✼

slide-41
SLIDE 41

❊①❛♠♣❧❡ ♦❢ ❛ ♣r♦♣❡rt② ❢♦r t②♣❡ ✷ str✉❝t✉r❡s ❆ ❣r❛♣❤ ✐s s✐♠♣❧❡ ✿ ∀e1 ∀e2 ∀x ∀y ⋁

a∈Σ

((❡❞❣a(e1,x,y) ∧ ❡❞❣a(e2,x,y)) ⇒ e1 = e2) ❛ss✉♠✐♥❣ t❤❛t α(Σ) = {2}

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✹ ✴ ✾✼

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SLIDE 42

▲✐♠✐t❛t✐♦♥s ♦❢ ❋❖

■♠♣♦ss✐❜❧❡ t♦ ❡①♣r❡ss t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♣❛t❤ ❢r♦♠ ♦♥❡ ✈❡rt❡① t♦ ❛♥♦t❤❡r ■♥ ❢❛❝t✱ ✐♥ ❛ ❝❡rt❛✐♥ s❡♥s❡ ♦♥❧② ❧♦❝❛❧ ♣r♦♣❡rt✐❡s ♠❛② ❜❡ ❡①♣r❡ss❡❞✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✺ ✴ ✾✼

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SLIDE 43

❍❛♥❢✬s✴●❛✐❢♠❛♥✬s✲❧✐❦❡ t❤❡♦r❡♠

r✲❧♦❝❛❧ ❢♦r♠✉❧❛ ✭❛r♦✉♥❞ x✮ ✿ q✉❛♥t✐✜❝❛t✐♦♥ ✐s ❜♦✉♥❞ t♦ ❛♥ r✲♥❡✐❣❤❜♦✉r❤♦♦❞ ∃y (δ(x,y) ≤ r ∧ ϕ) ♦r ∀y (δ(x,y) ≤ r ⇒ ϕ) ✇❤❡r❡ ✏δ(x,y) ≤ r✑ ✐s ❛ ✶st ♦r❞❡r ❢♦r♠✉❧❛✱ ϕ ✐s ❡✐t❤❡r q✉❛♥t✐✜❡r✲❢r❡❡ ♦r r✲❧♦❝❛❧ ❛r♦✉♥❞ x

❚❤❡♦r❡♠ ✭❙❝❤✇❡♥t✐❝❦ ❛♥❞ ❇❛rt❤❡❧♠❛♥♥✱ ✶✾✾✽✮

❊✈❡r② ✜rst ♦r❞❡r ❢♦r♠✉❧❛ ✐s ❡q✉✐✈❛❧❡♥t t♦ ∃x ∀y ϕ(x,y) ✇❤❡r❡ ϕ ✐s r✲❧♦❝❛❧ ❛r♦✉♥❞ x ❢♦r s♦♠❡ r ∈ IN✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✻ ✴ ✾✼

slide-44
SLIDE 44

❋❖✭❘✮ ✿ ✜rst ♦r❞❡r ❧♦❣✐❝ ✇✐t❤ r❡❛❝❤❛❜✐❧✐t②

❋❖ ✰ r❡❛❝❤❛❜✐❧✐t② ♣r❡❞✐❝❛t❡ r❡❛❝❤(x,y) ♣❛t❤ ✇✐t❤ ❧❛❜❡❧ w ∈ Σ∗ ✐♥ A ❢r♦♠ d ∈ A t♦ e ∈ A ✐s ❞❡✜♥❡❞ ✐♥❞✉❝t✐✈❡❧② ✭0✮ w = ε ✇❤❡♥ d = e ✭n + 1✮ w = ua ✇❤❡♥ t❤❡r❡ ✐s ❛ ♣❛t❤ ✇✐t❤ ❧❛❜❡❧ u ❢r♦♠ d t♦ s♦♠❡ d′ ∈ A s✳t✳ e = d′ ✇❤❡♥ α(a) = 1 ❛♥❞ e ∈ A(a) ♣r♦✈✐❞❡❞ t❤❛t a ✐s ♥♦t t❤❡ ❧❛st ❧❡tt❡r ♦❢ u✱ ✇❤❡♥ α(a) > 1 t❤❡r❡ ❡①✐sts (d1,...,dn) ∈ A(a) s✳t✳ d′ = di✱ e = dj ❢♦r s♦♠❡ i < j✳ A ⊧ r❡❛❝❤(d,e) t❤❡r❡ ✐s ❛ ♣❛t❤ ❢r♦♠ d t♦ e✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✼ ✴ ✾✼

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SLIDE 45

❊①❛♠♣❧❡ ✭r❡❛❝❤❛❜✐❧✐t②✮

1 2 a 3 4 5

A

b d d e

α(a)=3 α(b) =4 α(c)=1 α(d)=α(e)=2 A ⊧ r❡❛❝❤(0,5) s♦♠❡ ♣❛t❤ ❧❛❜❡❧s ✿ dacb✱ eedacb✱ daddacb✱ dacbacb ❊①❛♠♣❧❡ ✭♣r♦♣❡rt②✮ ◆♦ ❞❡❛❞❧♦❝❦ ♠❛② ♦❝❝✉r ✿ ∀x ∀y ((ι(x) ∧ r❡❛❝❤(x,y)) ⇒ ∃z (z ≠ y ∧ ⋁

a∈Σ

a(y,z)))

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✽ ✴ ✾✼

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SLIDE 46

❋❖✭❘✱ s✉❜✮ ✿ ❛ s❧✐❣❤t ❡①t❡♥s✐♦♥ ♦❢ ❋❖✭❘✮

❋❖ ✰ r❡❛❝❤❛❜✐❧✐t② ♣r❡❞✐❝❛t❡s ✿ r❡❛❝❤Γ(x,y) ❢♦r ❡✈❡r② Γ ⊆ Σ A ⊧ r❡❛❝❤Γ(d,e) t❤❡r❡ ✐s ❛ ♣❛t❤ ❢r♦♠ d t♦ e ❧❛❜❡❧❧❡❞ ❜② ❛ ✇♦r❞ ✐♥ Γ∗✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✾ ✴ ✾✼

slide-47
SLIDE 47

❊①❛♠♣❧❡ ✭♣r♦♣❡rt②✮ ❆♥ ♦✛✲❧✐♥❡ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ♦✈❡r ❛♥ ✐♥♣✉t ❛❧♣❤❛❜❡t ∆ ❛ ✇♦r❦ t❛♣❡ ✇✐t❤ r❡❛❞✴✇r✐t❡ ❤❡❛❞✱ ❛♥ ✐♥♣✉t t❛♣❡ ✇✐t❤ r❡❛❞✲♦♥❧② ❤❡❛❞ t❤❛t ❝❛♥♥♦t ♠♦✈❡ ❜❛❝❦✇❛r❞s✳ ❚❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ ❚▼ ✐s r❡♣r❡s❡♥t❡❞ ❜② ✐ts tr❛♥s✐t✐♦♥ ❣r❛♣❤ ✈❡rt✐t❡s ❛r❡ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥s ♦❢ ❚▼ ❡✈❡r② ❡❞❣❡ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠♦✈❡ ❛♥ ❡❞❣❡ ✐s ❧❛❜❡❧❧❡❞ ❜② a ∈ ∆ ✐❢ a ✐s r❡❛❞ ♦♥ t❤❡ ✐♥♣✉t t❛♣❡ ❛♥ ❡❞❣❡ ✐s ❧❛❜❡❧❧❡❞ ❜② τ ✐❢ ❚▼ ♣❡r❢♦r♠s ❛ s✐❧❡♥t ♠♦✈❡ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ❡①♣r❡ss❡s t❤❡ ♣r♦♣❡rt② t❤❛t ❛ ❚▼ ❞♦❡s ♥♦t ❧♦♦♣✳ ∀x ∀y ((ι(x)∧r❡❛❝❤(x,y)) ⇒ ¬∃z (z ≠ y∧r❡❛❝❤{τ}(y,z)∧r❡❛❝❤{τ}(z,y)))

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✵ ✴ ✾✼

slide-48
SLIDE 48

▼♦r❡ ❣❡♥❡r❛❧ ❡①t❡♥s✐♦♥s ♦❢ ❋❖✭❘✮

▲❡t L ⊆ ℘(Σ∗) ❜❡ ❛ ❧❛♥❣✉❛❣❡ ❢❛♠✐❧② ❋❖✭L✮ ✿ ❋❖ ✰ r❡❛❝❤❛❜✐❧✐t② ♣r❡❞✐❝❛t❡s r❡❛❝❤L(x,y) ❢♦r ❡✈❡r② ❧❛♥❣✉❛❣❡ L ∈ L A ⊧ r❡❛❝❤L(d,e) ✇❤❡♥ t❤❡r❡ ✐s ❛ ♣❛t❤ ❢r♦♠ d t♦ e ❧❛❜❡❧❧❡❞ ❜② ❛ ✇♦r❞ ✐♥ L✳ ▼❛✐♥❧② ❋❖✭❘❡❣✮ ❢♦r t❤❡ ❢❛♠✐❧② ❘❡❣ ♦❢ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s ❋❖✭❈❋✮ ❢♦r t❤❡ ❢❛♠✐❧② ❈❋ ♦❢ ❝♦♥t❡①t✲❢r❡❡ ❧❛♥❣✉❛❣❡s

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✶ ✴ ✾✼

slide-49
SLIDE 49

❋❖✭❚❈✮ ✿ ✜rst ♦r❞❡r ❧♦❣✐❝ ✇✐t❤ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡

❋❖ ✰ ❢♦r♠✉❧❛s [TCx,yϕ(x,y,z)](s,t) ✇❤❡r❡ x,y,z ❛r❡ t✉♣❧❡s ♦❢ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ✈❛r✐❛❜❧❡s s,t ❛r❡ t✉♣❧❡s ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ ∣x∣ = ∣y∣ = ∣s∣ = ∣t∣ = n ❢r❡❡(ϕ(x,y,z)) ❂ {x,y,z} ❢r❡❡([TCx,yϕ(x,y,z)](s,t)) ❂ {s,t,z} (A,d1,d2,d3) ⊧ [TCx,yϕ(x,y,z)](s,t) ✇❤❡♥ (d1,d2) ❜❡❧♦♥❣s t♦ t❤❡ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ♦❢ t❤❡ r❡❧❛t✐♦♥ {(e1,e2) ∣ (A,e1,e2,d3) ⊧ ϕ(x,y,z)} ❋❖✭❚❈✮(n) ✿ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ❛♣♣❧✐❡s t♦ r❡❧❛t✐♦♥s ♦♥ t✉♣❧❡s ♦❢ ❧❡♥❣t❤ ❛t ♠♦st n✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✷ ✴ ✾✼

slide-50
SLIDE 50

❊①❛♠♣❧❡ ♦❢ ❋❖✭❚❈✮(1) ❢♦r♠✉❧❛ ❋♦r ❛ ❧❛♥❣✉❛❣❡ ❣✐✈❡♥ ❜② ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ R✱ r❡❛❝❤R(x,y) ✐s ❞❡✜♥❡❞ ✐♥❞✉❝t✐✈❡❧② r❡❛❝❤R(x,y) ∶⇔ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x = y ✐❢ R = ε a(x,y) ✐❢ R = a r❡❛❝❤R1(x,y) ∨ r❡❛❝❤R2(x,y) ✐❢ R = R1 + R2 ∃z (r❡❛❝❤R1(x,z)r❡❛❝❤R2(z,y), ✐❢ R = R1R2 [❚❈str❡❛❝❤P (s,t)](x,y) ✐❢ R = P ∗ ❊①❛♠♣❧❡ ♦❢ ❋❖✭❚❈✮(2) ❢♦r♠✉❧❛ ❋♦r tr❡❡s ✿ t✇♦ ♥♦❞❡s x ❛♥❞ y ❛r❡ ❛t t❤❡ s❛♠❡ ❞❡♣t❤ ✿ ∃r (∀z z ↛ r ∧ [❚❈x1y1,x2y2(x1 → x2 ∧ y1 → y2)](rr,xy))

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✸ ✴ ✾✼

slide-51
SLIDE 51

❙❖ ✿ ✷♥❞ ♦r❞❡r ❧♦❣✐❝

✶st ♦r❞❡r ✈❛r✐❛❜❧❡s ✿ x,y,z,... r❛♥❣❡ ♦✈❡r ❡❧❡♠❡♥ts ♦❢ A ✷♥❞ ♦r❞❡r ✈❛r✐❛❜❧❡s ✿ X,Y,Z,... r❛♥❣❡ ♦✈❡r r❡❧❛t✐♦♥s ♦♥ A ❡✈❡r② ✈❛r✐❛❜❧❡ X ❤❛s ✐ts ❛r✐t② α(X) ❙❖ ✐s t❤❡ ✉s✉❛❧ ❧♦❣✐❝ ❢♦r ♠❛t❤❡♠❛t✐❝s✳ ❊①❛♠♣❧❡ ♦❢ ❛ ❞❡✜♥❛❜❧❡ r❡❧❛t✐♦♥ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ eqp(X,Y ) s❛②s t❤❛t s❡ts X ❛♥❞ Y ❛r❡ ❡q✉✐♣♦t❡♥t✱ ∃Z ( ∀x (X(x) ⇒ ∃y (Y (y) ∧ Z(x,y)) ∧ ∀y (Y (y) ⇒ ∃x (X(x) ∧ Z(x,y)) ∧ ∀x ∀y1 ∀y2 ((Z(x,y1) ∧ Z(x,y2)) ⇒ y1 = y2) ∧ ∀x1 ∀x2 ∀y ((Z(x1,y) ∧ Z(x2,y)) ⇒ x1 = x2) ) ✇❤❡r❡ α(X) = α(Y ) = 1, α(Z) = 2✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✹ ✴ ✾✼

slide-52
SLIDE 52

▼❙❖ ✿ ♠♦♥❛❞✐❝ ✷♥❞ ♦r❞❡r ❧♦❣✐❝

▼❙❖1 ✶st ♦r❞❡r ✈❛r✐❛❜❧❡s ✿ x,y,z,... r❛♥❣❡ ♦✈❡r ❡❧❡♠❡♥ts ♦❢ A ✷♥❞ ♦r❞❡r ✈❛r✐❛❜❧❡s ✿ X,Y,Z,... r❛♥❣❡ ♦✈❡r s✉❜s❡ts ♦❢ A ✈✐③✳✱ α(X) = 1 ❢♦r ❡✈❡r② X ▼❙❖2 ❡❛❝❤ x r❛♥❣❡s ❡✐t❤❡r ♦✈❡r ❡❧❡♠❡♥ts ♦❢ A✈❡rt ♦r ♦✈❡r ❡❧❡♠❡♥ts ♦❢ A❡❞❣ ❡❛❝❤ X r❛♥❣❡s ❡✐t❤❡r ♦✈❡r s✉❜s❡ts ♦❢ A✈❡rt ♦r ♦✈❡r s✉❜s❡ts ♦❢ A❡❞❣ ▼❙❖2 ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦ ❝❛❧❧❡❞ ❣✉❛r❞❡❞ ✷♥❞ ♦r❞❡r ❧♦❣✐❝ ✭●❙❖✮✳ ❊①t❡♥s✐♦♥s ✿ ▼❙❖ω ∣X∣ ≥ ℵ0 ▼❙❖❝ ∣X∣ ✪ m = n ❢♦r m,n ∈ IN ▼❙❖ ✇✐t❤ ❝♦✉♥t✐♥❣ ❘❡str✐❝t✐♦♥s ✿ ❲▼❙❖ ❡✈❡r② X r❛♥❣❡s ♦✈❡r ✜♥✐t❡ s❡ts ✇❡❛❦ ▼❙❖

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✺ ✴ ✾✼

slide-53
SLIDE 53

❊①❛♠♣❧❡ ❢♦r t②♣❡ ✶ str✉❝t✉r❡s

❆ ❞❡t❡r♠✐♥✐st✐❝ ❚▼ ❤❛❧ts ♦♥ ❡✈❡r② ✐♥♣✉t ✿ ∀x (ι(x) ⇒ (∃y r❡❛❝❤(x,y) ∧ ∀z y ↛ z)) ✇❤❡r❡ r❡❛❝❤(x,y) ✐s ∀X ⎛ ⎝(x ∈ X ∧ ∀z1 ∀z2 ((z1 ∈ X ∧ z1 → z2) ⇒ z2 ∈ X)) ⇒ y ∈ X⎞ ⎠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✻ ✴ ✾✼

slide-54
SLIDE 54

❊①❛♠♣❧❡ ❢♦r t②♣❡ ✷ str✉❝t✉r❡s ❆ ❣r❛♣❤ ✭✇✐t❤ ❛t ❧❡❛st ✷ ✈❡rt✐❝❡s✮ ❤❛s ❛ ❍❛♠✐❧t♦♥✐❛♥ ❝✐r❝✉✐t ✿ ∃E ∀x ⎛ ⎝∃e1 ∃e2 ∃y ∃z (e1 ≠ e2 ∧ e1 ∈ E ∧ e2 ∈ E ∧ ❡❞❣(e1,y,x) ∧ ❡❞❣(e2,x,z) ∧ ∀e3 ∀y (e3 ∈ E ⇒ ((❡❞❣(e3,y,x) ⇒ e3 = e1) ∧ (❡❞❣(e3,x,y) ⇒ e3 = e2)))) ∧ s❝(E)⎞ ⎠ ❊✈❡r② ✈❡rt❡① ❤❛s ❡①❛❝t❧② ♦♥❡ ✐♥❝♦♠✐♥❣ ❡❣❡ ✐♥ E ❛♥❞ ♦✉t❣♦✐♥❣ ❡❞❣❡ ✐♥ E ❛♥❞ t❤❡ ❣r❛♣❤ ✐s str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❜② E✳ ❋♦r♠✉❧❛ s❝(E) ✉s❡❞ ❛❜♦✈❡ ✐s ∀x ∀y ∀X ⎛ ⎝(x ∈ X ∧ ∀z1 ∀z2 ∀e ((z1 ∈ X ∧ e ∈ E ∧ ❡❞❣(e,z1,z2)) ⇒ z2 ∈ X)) ⇒ y ∈ X⎞ ⎠

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✼ ✴ ✾✼

slide-55
SLIDE 55

▲♦❣✐❝ ✿ s✉♠♠❛r②

F O F O(R)

F O(R, sub)

F O(Reg)

F O(CF)

F O(TC)(1) ⊊

⋯ ⊊

F O(TC)(n) ⊊

⋯ ⊊ ⋯

F O(TC)

WMSO ⊊ MSO1 ⊊ MSO2 ⊊ MSOω

2 ⊊

MSO

2

⊊ ⊊

MSOω

1

MSO

1

⊊ ⊊ ⊊

SO

⊊ ⊊ ⊊

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✽ ✴ ✾✼

slide-56
SLIDE 56

❙tr✉❝t✉r❡ ❜✉✐❧❞✐♥❣ ♦♣❡r❛t✐♦♥s

❝❧❛ss ♦❢ str✉❝t✉r❡s S1 S2 ❧♦❣✐❝ L1 L2 ❙tr✉❝t✉r❡ ❜✉✐❧❞✐♥❣ ♦♣❡r❛t✐♦♥ f∶S1 → S2 ✇✐t❤ ❜❛❝❦✇❛r❞s tr❛♥s❧❛t✐♦♥ ♣r♦♣❡rt② ✿ ❋♦r ❡✈❡r② str✉❝t✉r❡ A ∈ S1 ❛♥❞ ❢♦r ❡✈❡r② s❡♥t❡♥❝❡ ϕ ∈ L2 A ⊧ f♯(ϕ) ⇔ f(A) ⊧ ϕ ✇❤❡r❡ f♯∶L2 → L1 ✐s ❡✛❡❝t✐✈❡✳ ❘❡♠❛r❦ ■❢ ❚❤L1(A) ✐s r❡❝✉rs✐✈❡✱ s♦ ✐s ❚❤L2(f(A))✳ f ✐s (L1,L2)✲❝♦♠♣❛t✐❜❧❡

  • ❡♥❡r❛❧ ❝❛s❡ ✿

f(A1,...,An)

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✾ ✴ ✾✼

slide-57
SLIDE 57

▼❙❖ ✐♥t❡r♣r❡t❛t✐♦♥s

A ∈ Struct[Σ]

d

↦ B ∈ Struct[Γ] ❞❡✜♥✐t✐♦♥ s❝❤❡♠❡ ✭♣❛r❛♠❡t❡r❧❡ss✮ d = ⟨δ(x),(θb(x1,...,xα(b)))b∈Γ,⟩ ✇❤❡r❡ δ(x) ∈ Form▼❙❖[Σ]✱ ✭❞❡✜♥❡s ✉♥✐✈❡rs❡ B ♦❢ B✮ θb(x1,...,xα(b)) ∈ Form▼❙❖[Σ]✱ ✭❞❡✜♥❡s B(b)✮ B = d(A) ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ✿ B ∶= {d ∈ A ∣ A ⊧ δ(d)} B(b) ∶= {d ∣ A ⊧ θb(d)} ❢♦r ❡✈❡r② b ∈ Γ

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✵ ✴ ✾✼

slide-58
SLIDE 58

❇❛❝❦✇❛r❞s tr❛♥s❧❛t✐♦♥

d♯∶ Form▼❙❖[Γ] → Form▼❙❖[Σ] d♯(x = y) = x = y d♯(x ∈ X) = x ∈ X d♯(b(x1,...,xα(b))) = θb(x1,...,xα(b)) d♯(¬ϕ) = ¬d♯(ϕ) d♯(ϕ1 ∨ ϕ2) = d♯(ϕ1) ∨ d♯(ϕ2) d♯(∃x ϕ) = ∃x(δ(x) ∧ d♯(ϕ)) d♯(∃X ϕ) = ∃X (∀x (x ∈ X ⇒ δ(x)) ∧ d♯(ϕ)) ❆♥ ▼❙❖ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ▼❙❖✲❝♦♠♣❛t✐❜❧❡ ✭✈✐③✳ ✭▼❙❖✱ ▼❙❖✮✲❝♦♠♣❛t✐❜❧❡✮✳ ■❢ A ❤❛s ❛ ❞❡❝✐❞❛❜❧❡ ▼❙❖ t❤❡♦r②✱ t❤❡♥ d(A) ❤❛s ❛ ❞❡❝✐❞❛❜❧❡ ▼❙❖ t❤❡♦r②✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✶ ✴ ✾✼

slide-59
SLIDE 59

❊①❛♠♣❧❡

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

l r l r l r l r l r l r l r

Σ = {l,r} α(l) = α(r) = 2 Γ = {a,b,c} α(a) = α(b) = α(c) = 2 d = ⟨δ(x),θa(x,y),θb(x,y),θc(x,y)⟩ δ(x) = ∃t (¬∃z (z

l

  • → t ∨ z

r

  • → t) ∧

∀X ((t ∈ X ∧ ∀z1 ∀z2 ((z1 ∈ X ∧ z1

l

  • → z2) ⇒ z2 ∈ X))⇒

(x ∈ X ∨ ∃z (z ∈ X ∧ z

r

  • → x))))

θa(x,y) = x

l

  • → y

θb(x,y) = ∃z1 ∃z2 (z1

l

  • → z2 ∧ z1

r

  • → y ∧ z2

r

  • → x)

z1 z2 y x

l r r

θc(x,y) = x

r

  • → y

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✷ ✴ ✾✼

slide-60
SLIDE 60 10 11 010 011 100 101 110 111

ε 1 00 01 000 001 l

r

l

r

l

r

l

r

l

r

l

r

l

r

a a a a a a a c c c c c c c b b b

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✸ ✴ ✾✼

slide-61
SLIDE 61

ε 1 00 01 000 001 a a a c c c b b

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✹ ✴ ✾✼

slide-62
SLIDE 62

ε 1 01 00 001 000 0001 a a a c c c c b b b

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✺ ✴ ✾✼

slide-63
SLIDE 63

▼❙❖ tr❛♥s❞✉❝t✐♦♥s

A ∈ Struct[Σ]

d

↦ B ∈ Struct[Γ] ❞❡✜♥✐t✐♦♥ s❝❤❡♠❡ ✭♣❛r❛♠❡t❡r❧❡ss✮ d = ⟨n,(δi(x))i∈[n],(θb,k(x1,...,xα(b)))b∈Γ,k∈[n]α(b)⟩ ✇❤❡r❡ n ∈ IN✱ ✭♥✉♠❜❡r ♦❢ ❝♦♣✐❡s✮ δi(x) ∈ Form▼❙❖[Σ]✱ ✭❞❡✜♥❡ ✉♥✐✈❡rs❡ B ♦❢ B✮ θb,k(x1,...,xα(b)) ∈ Form▼❙❖[Σ]✱ ✭❞❡✜♥❡ B(b)✮ B = d(A) ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ✿ B ∶= ⋃

i∈[n]

{(d,i) ∣ A ⊧ δi(d)} B(b) ∶= ⋃

i1...iα(b)∈[n]α(b)

{((d1,i1),...,(dα(b),iα(b))) ∣ A ⊧ θb,i1...iα(b)(d1,...,dα(b))} ❢♦r ❡✈❡r② b ∈ Γ

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✻ ✴ ✾✼

slide-64
SLIDE 64

❊①❛♠♣❧❡

d ♠❛♣s ❛♥② ✇♦r❞ u ∈ {a,b}∗ t♦ u3 Σ = {s,a,b} α(s) = 1 α(a) = α(b) = 1 d = ⟨3,δ1(x)✱ δ2(x),δ3(x),θs,11(x,y),θs,12(x,y),θs,13(x,y), θs,21(x,y),θs,22(x,y),θs,23(x,y),θs,31(x,y), θs,32(x,y),θs,33(x,y),θa,1(x),θa,2(x),θa,3(x), θb,1(x),θb,2(x),θb,2(x)⟩ a

t

b

u

a

v

s s a

t1

b

u1

a

v1

a

t2

b

u2

a

v2

a

t3

b

u3

a

v3

s s s s s s s s d δ1(x) = δ2(x) = δ3(x) = tr✉❡ θa,1(x)=θa,2(x)=θa,3(x)=a(x) θs,11(x,y) = θs,22(x,y) = θs,33(x,y) = s(x) θb,1(x) = θb,2(x) = θb,3(x) = b(x) θs,21(x,y) = θs,32(x,y) = θs,31(x,y) = ❢❛❧s❡ θs,13(x,y) = ❢❛❧s❡ θs,12(x,y) = θs,23(x,y) = ¬∃z (s(x,z) ∨ s(z,y))

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✼ ✴ ✾✼

slide-65
SLIDE 65

❋❖ ✐♥t❡r♣r❡t❛t✐♦♥s

A ∈ Struct[Σ]

d

↦ B ∈ Struct[Γ] ❞❡✜♥✐t✐♦♥ s❝❤❡♠❡ ✭♣❛r❛♠❡t❡r❧❡ss✮ d = ⟨n,δ(x),(θb(x1,...,xα(b)))b∈Γ⟩ ✇❤❡r❡ n ∈ IN ✭✐s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ d✮✱ δ(x) ∈ FormL[Σ]✱ ✭❞❡✜♥❡s ✉♥✐✈❡rs❡ B ♦❢ B✮ ❡✈❡r② θb(x1,...,xα(b)) ∈ FormL[Σ]✱ ✭❞❡✜♥❡s B(b)✮ B = d(A) ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ✿ B ∶= {d ∈ An ∣ A ⊧ δ(d)} B(b) ∶= {(d1,...,dα(b)) ∣ A ⊧ θb(d1,...,dα(b))} ❢♦r ❡✈❡r② b ∈ Γ

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✽ ✴ ✾✼

slide-66
SLIDE 66

❆ss✉♠✐♥❣ t❤❛t ❢♦r♠✉❧❛s ♦❢ d ❛r❡ ✇r✐tt❡♥ ✐♥ ❛ ❧♦❣✐❝ L✱ d♯∶ Form❋❖[Γ] → FormL[Σ] ❋✐rst✱ t♦ ❡✈❡r② ✈❛r✐❛❜❧❡ x ✇❡ ❛ss♦❝✐❛t❡ ❛ t✉♣❧❡ ♦❢ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ♥❡✇ ✈❛r✐❛❜❧❡s x = x1 ...xn✳ ❚❤❡♥ ✿ d♯(x = y) = x1 = y1 ∧ ⋯ ∧ xn = yn d♯(b(x1,...,xα(b))) = θb(x1,...,xα(b)) d♯(¬ϕ) = ¬d♯(ϕ) d♯(ϕ1 ∨ ϕ2) = d♯(ϕ1) ∨ d♯(ϕ2) d♯(∃xϕ) = ∃x(δ(x) ∧ d♯(ϕ)) ❆♥ ❋❖ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s (L,❋❖)✲❝♦♠♣❛t✐❜❧❡✳ ■❢ A ❤❛s ❛ ❞❡❝✐❞❛❜❧❡ L t❤❡♦r②✱ t❤❡♥ d(A) ❤❛s ❛ ❞❡❝✐❞❛❜❧❡ ❋❖ t❤❡♦r②✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✾ ✴ ✾✼

slide-67
SLIDE 67

Pr♦❞✉❝ts

❲❡ ❛ss✉♠❡ t❤❛t max(α(Σi)) ≤ 2✱ ❢♦r i ∈ [n]✳ ✭■♥ t❤❡ ❝❛s❡ ♦❢ ❛r✐t② ❤✐❣❤❡r t❤❛♥ ✷ s❡✈❡r❛❧ ❞❡✜♥✐t✐♦♥s ♦❢ ♣r♦❞✉❝t ❛r❡ ♣♦ss✐❜❧❡✳✮ ▲❡t Ai ∈ Struct[Σi]✳ ❚❤❡♥ C = ∏

i∈[n]

Ai ✐s ❛ str✉❝t✉r❡ ♦✈❡r Γ✱ ✇❤❡r❡ ✿ Γ(1) ∶= ∏i∈[n](Σ(1)

i

∪ {tr✉❡}) ∖ {(tr✉❡,...,tr✉❡)} Γ(2) ∶= ∏i∈[n](Σ(2)

i

∪ {=}) ∖ {=,...,=)} C ∶= ∏i∈[n] Ai C(a) ∶= {d ∣ di ∈ Ai(ai) ❢♦r i ∈ [n]}, a ∈ Γ(1) C(b) ∶= {(d,d

′) ∣ (di,d′ i) ∈ Ai(bi) ❢♦r i ∈ [n]},

b ∈ Γ(2)

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✵ ✴ ✾✼

slide-68
SLIDE 68

Σ(1)

1

= {l} Σ(1)

2

= {a} Σ(2)

1

= {f,g} Σ(2)

2

= {b,c} Γ(1) = {(tr✉❡,a),(l,tr✉❡),(l,a)} Γ(2) = {(=,b),(=,c),(f,=),(f,b),(f,c),(g,=),(g,b),(g,c)} a

1

b c l

u v

f g

u0

(l,

true)

u1

(l,a)

v0 v1

(

true,a)

(=,b) (=,b) (f,=) (f,c) (g,=) (g,c) (f,=) (f,c) (g,=) (g,c) (f,b) (g,b) c c

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✶ ✴ ✾✼

slide-69
SLIDE 69

❙②♥❝❤r♦♥✐③❡❞ ♣r♦❞✉❝t

❙②♥❝❤r♦♥✐③❛t✐♦♥ ❝♦♥str❛✐♥t ∆ ⊆ Γ ❊①❛♠♣❧❡ ∆ = {(l,a),(f,b),(f,c),(g,b),(g,c)} a

1

b c l

u v

f g

u0 u1

(l,a)

v0 v1

(f,c) (g,c) (f,c) (g,c) (f,b) (g,b)

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✷ ✴ ✾✼

slide-70
SLIDE 70

Pr♦❞✉❝t ✐s ❋❖✲❝♦♠♣❛t✐❜❧❡

❚❤❡♦r❡♠ ✭▼♦st♦✇s❦✐✱ ✶✾✺✷✮

▲❡t C = A × B✳ ❚❤❡♥ ❚❤❋❖(C) ✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜②✱ ❛♥❞ ♠❛② ❜❡ ❡✛❡❝t✐✈❡❧② ❝♦♠♣✉t❡❞ ❢r♦♠ ❚❤❋❖(A) ❛♥❞ ❚❤❋❖(B)✳ ■♥ ✶✾✺✾ ❋❡❢❡r♠❛♥ ❛♥❞ ❱❛✉❣❤t ❣❛✈❡ ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ❢♦r ✐♥✜♥✐t❡ ♣r♦❞✉❝ts ∏i∈I Ai✳ ❇✉t ❚❤❋❖✭❘✮(A) ❛♥❞ ❚❤❋❖✭❘✮(B) ❞❡❝✐❞❛❜❧❡ ⇏ ❚❤❋❖✭❘✮(A × B) ❞❡❝✐❞❛❜❧❡ ❋♦r ✷✲❞✐♠❡♥s✐♦♥❛❧ ✐♥✜♥✐t❡ ❣r✐❞ G2 = ⟨IN,s✉❝⟩ × ⟨IN,s✉❝⟩ ❚❤❋❖✭❘✮(G2) ✭❤❡♥❝❡ ❚❤▼❙❖(G2)✮ ✐s ♥♦t r❡❝✉rs✐✈❡✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✸ ✴ ✾✼

slide-71
SLIDE 71 1

s

2

s

3

s

1

s

2

s

3

s

(0,0) (1,0)

s1

(2,0)

s1

(3,0)

s1

(0,1) (1,1)

s1

(2,1)

s1

(3,1)

s1

(0,2) (1,2)

s1

(2,2)

s1

(3,2)

s1

(0,3) (1,3)

s1

(2,3)

s1

(3,3)

s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 s2 ❙②♥❝❤r♦♥✐s❛t✐♦♥ ❝♦♥str❛✐♥t ∆ = {s1,s2} ✇❤❡r❡ s1 = (s,=) s2 = (=,s)

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✹ ✴ ✾✼

slide-72
SLIDE 72

❉✐s❥♦✐♥t s✉♠

▲❡t Ai ∈ Struct[Σi]✳ ❚❤❡♥ C = ∐

i∈[n]

Ai ✐s ❛ str✉❝t✉r❡ ♦✈❡r Γ = ⋃i∈[n] Σi✱ ✇❤❡r❡ ✿ C ∶= ⋃i∈[n] Ai × {i} C(a) ∶= {((d1,i),...,(dk,i)) ∣ (d1,...,dk) ∈ Ai(a)} ❢♦r a ∈ Σi

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✺ ✴ ✾✼

slide-73
SLIDE 73

❊①❛♠♣❧❡

u

a

v

b c l

u v

f g l

u1 v1

f gu2 a

v2

b c

❚❤❡♦r❡♠ ✭❙❤❡❧❛❤✱ ✶✾✼✺✮

❉✐s❥♦✐♥t s✉♠ ✐s ▼❙❖✲❝♦♠♣❛t✐❜❧❡

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✻ ✴ ✾✼

slide-74
SLIDE 74

❯♥❢♦❧❞✐♥❣

▲❡t A ∈ Struct[Σ] ❜❡ ❛ ❣r❛♣❤ ✭max(α(Σ)) = 2✮ ❛♥❞ r ∈ A✳ ❚❤❡ ✉♥❢♦❧❞✐♥❣ C = ❯♥(r,A) ♦❢ A ❢r♦♠ r ❜❡❧♦♥❣s t♦ Struct[Σ] ❛♥❞ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s C ∶= {d0a1 ...andn ∣ d0 = r, n ∈ IN, di−1

ai

  • → di ❢♦r i ∈ [n]}

C(a) ∶= { {(u,uad) ∣ u ∈ C, uad ∈ C, a ∈ Σ(2), d ∈ A} α(a) = 2 {ubd ∈ C ∣ u ∈ C, b ∈ Σ(2), d ∈ A(a)} α(a) = 1

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✼ ✴ ✾✼

slide-75
SLIDE 75

❊①❛♠♣❧❡

  • a

a a b b b b b b b

  • a

a a b b b b b b b b b b

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✽ ✴ ✾✼

slide-76
SLIDE 76

❚❤❡♦r❡♠ ✭❈♦✉r❝❡❧❧❡✱ ❲❛❧✉❦✐❡✇✐❝③✱ ✶✾✾✺✮

❚❤❡ ✉♥❢♦❧❞✐♥❣ ✐s ▼❙❖✲❝♦♠♣❛t✐❜❧❡ ❋♦r ❡✈❡r② ❣r❛♣❤ A ❛♥❞ r ∈ A✱ ❢♦r ❡✈❡r② ▼❙❖ s❡♥t❡♥❝❡ ψ ♦♥ ♠❛② ❝♦♥str✉❝t ❛♥ ▼❙❖ s❡♥t❡♥❝❡ ϕ s✉❝❤ t❤❛t A ⊧ ϕ ⇔ ❯♥(r,A) ⊧ ψ

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✾ ✴ ✾✼

slide-77
SLIDE 77

◗✉❡st✐♦♥ ✿ ❲❤✐❝❤ ♦❢ t❤❡ ✷ str✉❝t✉r❡s ✐s s✐♠♣❧❡r ❄

  • a

a a b b b b b b b

  • a

a a b b b b b b b b b b

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✵ ✴ ✾✼

slide-78
SLIDE 78

❆♥s✇❡r ✿ ▼❙❖ ✐♥t❡r♣r❡t❛t✐♦♥ ⟨δ(x),θa(x,y),θb(x,y)⟩

δ(x) = tr✉❡ θa(x,y) = x

a

  • → y

θb(x,y) = x

b

  • → y ∨ (¬∃z (x

a

  • → z ∨ x

b

  • → z)

∧ r❡❛❝❤ab∗(y,x))

anb2(n+1)

  • a

a a b b b b b b b b b b b b b

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✶ ✴ ✾✼

slide-79
SLIDE 79

⋅ l r ✉♥❢♦❧❞✐♥❣

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l r l r l r l r l r l r l r

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✷ ✴ ✾✼

slide-80
SLIDE 80

■t❡r❛t✐♦♥

▲❡t A ∈ Struct[Σ] ❚❤❡ ✐t❡r❛t✐♦♥ ♦❢ A ✐s ❛ str✉❝t✉r❡ A∗ ♦✈❡r Σ ∪ {♯,&} α(♯) = 2 α(&) = 1 A∗ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ✿ t❤❡ ✉♥✐✈❡rs❡ ♦❢ A∗ ✐s t❤❡ s❡t A∗ ♦❢ ❛❧❧ ✇♦r❞s ♦✈❡r A A∗(a) ∶= {(wd1,...,wdn) ∣ (d1,...,dn) ∈ A(a)} ❢♦r a ∈ Σ(n) A∗(♯) ∶= {(w,wd) ∣ w ∈ A∗,d ∈ A} A∗(&) ∶= {wdd ∣ w ∈ A∗,d ∈ A}

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✸ ✴ ✾✼

slide-81
SLIDE 81

❊①❛♠♣❧❡

Shelah-Stupp

⋅ ⋅ ⋅ 1 2 b a a

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

slide-82
SLIDE 82

❊①❛♠♣❧❡

Shelah-Stupp

⋅ ⋅ ⋅ 1 2 b a a 00 01 02 b a a ♯ ♯ ♯

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

slide-83
SLIDE 83

❊①❛♠♣❧❡

Shelah-Stupp

⋅ ⋅ ⋅ 1 2 b a a 00 01 02 b a a ♯ ♯ ♯ 10 11 12 b a a ♯ ♯ ♯

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

slide-84
SLIDE 84

❊①❛♠♣❧❡

Shelah-Stupp

⋅ ⋅ ⋅ 1 2 b a a 00 01 02 b a a ♯ ♯ ♯ 10 11 12 b a a ♯ ♯ ♯ 20 21 22 b a a ♯ ♯ ♯

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

slide-85
SLIDE 85

❊①❛♠♣❧❡

Shelah-Stupp

⋅ ⋅ ⋅ 1 2 b a a 00 01 02 b a a ♯ ♯ ♯ 10 11 12 b a a ♯ ♯ ♯ 20 21 22 b a a ♯ ♯ ♯ 000 001 002 b a a ♯ ♯ ♯

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

slide-86
SLIDE 86

❊①❛♠♣❧❡

Shelah-Stupp

⋅ ⋅ ⋅ 1 2 b a a 00 01 02 b a a ♯ ♯ ♯ 10 11 12 b a a ♯ ♯ ♯ 20 21 22 b a a ♯ ♯ ♯ 000 001 002 b a a ♯ ♯ ♯ 010 011 012 b a a ♯ ♯ ♯

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

slide-87
SLIDE 87

❊①❛♠♣❧❡

Shelah-Stupp

⋅ ⋅ ⋅ 1 2 b a a 00 01 02 b a a ♯ ♯ ♯ 10 11 12 b a a ♯ ♯ ♯ 20 21 22 b a a ♯ ♯ ♯ 000 001 002 b a a ♯ ♯ ♯ 010 011 012 b a a ♯ ♯ ♯ 020 021 022 b a a ♯ ♯ ♯

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

slide-88
SLIDE 88

❊①❛♠♣❧❡

Shelah-Stupp

⋅ ⋅ ⋅ 1 2 b a a 01 02 b a a ♯ ♯ ♯ 10 12 b a a ♯ ♯ ♯ 20 21 b a a ♯ ♯ ♯ 001 002 b a a ♯ ♯ ♯ 010 012 b a a ♯ ♯ ♯ 020 021 b a a ♯ ♯ ♯

Mu hnik &

00 11 22 000 011 022

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

slide-89
SLIDE 89

❚❤❡♦r❡♠ ✭▼✉❝❤♥✐❦✱ ✶✾✽✹✮

❚❤❡ ✉♥❢♦❧❞✐♥❣ ✐s ▼❙❖✲❝♦♠♣❛t✐❜❧❡ ❋♦r ❡✈❡r② ❣r❛♣❤ A ❛♥❞ ❡✈❡r② ▼❙❖ s❡♥t❡♥❝❡ ψ ♦♥ ♠❛② ❝♦♥str✉❝t ❛♥ ▼❙❖ s❡♥t❡♥❝❡ ϕ s✉❝❤ t❤❛t A ⊧ ϕ ⇔ A∗ ⊧ ψ ❋✉❧❧ ♣r♦♦❢ ❜② ❲❛❧✉❦✐❡✇✐❝③ ✐♥ ✶✾✾✻

Pr♦♣♦s✐t✐♦♥

❚❤❡ ✉♥❢♦❧❞✐♥❣ ❯♥(r,A) ❢r♦♠ ❛♥ ▼❙❖ ❞❡✜♥❛❜❧❡ r ∈ A ✐s ▼❙❖✲✐♥t❡r♣r❡t❛❜❧❡ ✐♥ A∗✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✺ ✴ ✾✼

slide-90
SLIDE 90

❙tr✉❝t✉r❡ ❜✉✐❧❞✐♥❣ ♦♣❡r❛t✐♦♥s ✿ s✉♠♠❛r②

♦♣❡r❛t✐♦♥ ❝♦♠♣❛t✐❜✐❧✐t② ▼❙❖ ✐♥t❡r♣r❡t❛t✐♦♥ ▼❙❖ ▼❙❖ tr❛♥s❞✉❝t✐♦♥ ▼❙❖ ❋❖ ✐♥t❡r♣r❡t❛t✐♦♥ ✭L,❋❖✮ ♣r♦❞✉❝t ❋❖ ❞✐s❥♦✐♥t s✉♠ ▼❙❖ ✉♥❢♦❧❞✐♥❣ ▼❙❖ ✐t❡r❛t✐♦♥ ▼❙❖

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✻ ✴ ✾✼

slide-91
SLIDE 91

❉❡s❝r✐❜✐♥❣ ❛♥ ✐♥✜♥✐t❡ str✉❝t✉r❡

❲❡ ✇❛♥t ❛ ✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥ t❤❛t ❝❛♥ ❜❡ ❝♦❞❡❞ ♦♥ ❛ ❝♦♠♣✉t❡r✳

  • ❧✐♥❡❛r str✉❝t✉r❡s
  • tr❡❡s

✜①♣♦✐♥t ❡q✉❛t✐♦♥s ❡✳❣✳ x = f(x,g(x))

  • ❣r❛♣❤s

r❡❧❛t✐♦♥s ❣r❛♣❤s✱ tr❛♥s✐t✐♦♥ ❣r❛♣❤s

  • ❛r❜✐tr❛r② str✉❝t✉r❡s

r❡❧❛t✐♦♥ ❣r❛♣❤s

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✼ ✴ ✾✼

slide-92
SLIDE 92

⋯ ←

a state register

qi

nite program instru tion 1 instru tion 2

instru tion n

⋯ ⋯

/ 1

10qi12

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✽ ✴ ✾✼

slide-93
SLIDE 93

❊①❛♠♣❧❡s ♦❢ ✭✐♥t❡r♥❛❧✮ ❝♦♥✜❣✉r❛t✐♦♥s 10q12 ❚❤❡r❡ ❛r❡ ♦♥❧② ❜❧❛♥❦s t♦ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦♥ t❤❡ r✐❣❤t ♦❢ ♦❢ 1012✳ ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ✷♥❞ 1✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s q✳ 1012p ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ✶st ❜❧❛♥❦ ❛❢t❡r 1012✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s p✳ 1012 r ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ✷♥❞ ❜❧❛♥❦ ❛❢t❡r 1012✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s r✳ q1012 ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ✶st 1✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s q✳ q 1012 ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ❜❧❛♥❦ ✐♠♠❡❞✐❛t❡❧② ❜❡❢♦r❡ 1012✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s q✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✾ ✴ ✾✼

slide-94
SLIDE 94

❊①❛♠♣❧❡s ♦❢ tr❛♥s✐t✐♦♥s 10q12

τ

  • → 1p022

❚❤❡ ❤❡❛❞ ♣r✐♥ts 2 ❛♥❞ ♠♦✈❡s ❧❡❢t✳ ◆♦ ✐♥♣✉t ✐s r❡❛❞ ✭s✐❧❡♥t tr❛♥s✐t✐♦♥✮✳ ❚❤❡ ♥❡✇ st❛t❡ ✐s p✳ 1012p

b

  • → 101q20

❚❤❡ ❤❡❛❞ ♣r✐♥ts 0 ❛♥❞ ♠♦✈❡s ❧❡❢t ✇❤✐❧❡ b ✐s r❡❛❞ ♦♥ t❤❡ ✐♥♣✉t t❛♣❡✳ ❚❤❡ ♥❡✇ st❛t❡ ✐s q✳ 101p2

a

  • → 10q1

❚❤❡ ❤❡❛❞ ♣r✐♥ts ❜❧❛♥❦ ❛♥❞ ♠♦✈❡s ❧❡❢t ✇❤✐❧❡ a ✐s r❡❛❞ ♦♥ t❤❡ ✐♥♣✉t t❛♣❡✳ ❚❤❡ ♥❡✇ st❛t❡ ✐s q✳ 1012 r

τ

  • → 1012 2p

❚❤❡ ❤❡❛❞ ♣r✐♥ts 2 ❛♥❞ ♠♦✈❡s r✐❣❤t✳ ◆♦ ✐♥♣✉t ✐s r❡❛❞✳ ❚❤❡ ♥❡✇ st❛t❡ ✐s p✳ q1012

a

  • → q

012 ❚❤❡ ❤❡❛❞ ♣r✐♥ts ❜❧❛♥❦ ❛♥❞ ♠♦✈❡s ❧❡❢t ✇❤✐❧❡ a ✐s r❡❛❞ ♦♥ t❤❡ ✐♥♣✉t t❛♣❡✳ ❚❤❡ st❛t❡ r❡♠❛✐♥s ✉♥❝❤❛♥❣❡❞✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✵ ✴ ✾✼

slide-95
SLIDE 95

❚r❛♥s✐t✐♦♥ ❣r❛♣❤s

✈❡rt✐❝❡s ✿ ✐♥t❡r♥❛❧ ❝♦♥✜❣✉r❛t✐♦♥s ❡❞❣❡s ✿ tr❛♥s✐t✐♦♥s ❜❡t✇❡❡♥ ❝♦♥✜❣✉r❛t✐♦♥s ✭st❛♥❞❛r❞ tr❛♥s✐t✐♦♥ ❣r❛♣❤✮ τ ∗✲❝❧♦s✉r❡s ♦❢ s✉❝❤ tr❛♥s✐t✐♦♥s ✭♦❜s❡r✈❛❜❧❡ tr❛♥s✐t✐♦♥ ❣r❛♣❤✮ M ♠❛❝❤✐♥❡ ✭❛✉t♦♠❛t♦♥✮ tgΣ(M) ✐ts tr❛♥s✐t✐♦♥ ❣r❛♣❤ tGΣ(M) ✐ts ♦❜s❡r✈❛❜❧❡ tr❛♥s✐t✐♦♥ ❣r❛♣❤ u1

a

u2

τ

u3

τ

u4

u1

a

  • a
  • a
  • u2

u3 u4 tgΣ(M) tGΣ(M)

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✶ ✴ ✾✼

slide-96
SLIDE 96

❘❡❧❛t✐♦♥ ❣r❛♣❤ ♦❢ ❛ ♠❛❝❤✐♥❡

✐♥♣✉t✴♦✉t♣✉t ♠❛❝❤✐♥❡s ✭❛✉t♦♠❛t❛✮

❄ ✲ ✛

q0 ❛ ❝ ... ❜ ❛ ❛

✲ ❄

❛ ❜ ... ❜ ❜ ❛ f ❜❡❣✐♥ ❡♥❞ ✐♥♣✉t ✇♦r❞ u ♦✉t♣✉t ✇♦r❞ v ♣❛✐r (u, v) ✐s ❛❝❝❡♣t❡❞ ❜② t❤❡ ♠❛❝❤✐♥❡

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✷ ✴ ✾✼

slide-97
SLIDE 97

❘❡❧❛t✐♦♥ ❣r❛♣❤ ♦❢ ❛ tr❛♥s❞✉❝❡r

✜♥✐t❡ tr❛♥s❞✉❝❡r

✻ ❄ ✛ ✛

q0 ✇r✐t❡ ❤❡❛❞ r❡❛❞ ❤❡❛❞ ... ❛ ❝ ... ❛ ❛ ❛ ❜ ❝

✻ ❄ ✲

f ✇r✐t❡ ❤❡❛❞ r❡❛❞ ❤❡❛❞ u v ♣❛✐r (u, v) ✐s ❛❝❝❡♣t❡❞ ❜② t❤❡ tr❛♥s❞✉❝❡r

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✸ ✴ ✾✼

slide-98
SLIDE 98

❘❡❧❛t✐♦♥ ❣r❛♣❤s

(u, v) ↦ u

a

  • → v

✶ ❢❛♠✐❧② ♦❢ ♠❛❝❤✐♥❡s (Ma ∣ a ∈ Σ) ✭r❡s♣✳ tr❛♥s❞✉❝❡rs (Ta ∣ a ∈ Σ)✮ ✷ ♠❛❝❤✐♥❡ ✭r❡s♣✳ tr❛♥s❞✉❝❡r✮ ✇✐t❤ ✜♥❛❧ st❛t❡s ❧❛❜❡❧❧❡❞ ✐♥ Σ

tgΣ(M) tgΣ(T)

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✹ ✴ ✾✼

slide-99
SLIDE 99

❘❡❧❛t✐♦♥ ❛♥❞ tr❛♥s✐t✐♦♥ ❣r❛♣❤s ✿ s✉♠♠❛r②

♦❜s❡r✈❛❜❧❡ tr❛♥s✐t✐♦♥ ❣r❛♣❤s ♠❡♠♦r② r❡str✐❝t✐♦♥ ♠❛❝❤✐♥❡ ❣r❛♣❤ ✐♥✜♥✐t❡✱ ♥♦ r❡str✐❝t✐♦♥ ❚✉r✐♥❣ r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ ♣✉s❤❞♦✇♥ st♦r❡ ♣✉s❤❞♦✇♥ ♣✉s❤❞♦✇♥ ✜♥✐t❡ ✜♥✐t❡ ✜♥✐t❡ r❡❧❛t✐♦♥ ❣r❛♣❤s ♠❛❝❤✐♥❡ ❣r❛♣❤ ❚✉r✐♥❣ r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡ ❧✐♥❛✐r❧② ❜♦✉♥❞❡❞ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ tr❛♥s❞✉❝❡r r❛t✐♦♥❛❧ s②♥❝❤r♦♥✐s❡❞ tr❛♥s❞✉❝❡r ❛✉t♦♠❛t✐❝

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✺ ✴ ✾✼

slide-100
SLIDE 100
  • r❛♣❤s ♦❢ ❧❛❜❡❧❧❡❞ str✐♥❣ r❡✇r✐t✐♥❣ s②st❡♠s

L = {li

a

  • → ri ∣ li,ri ∈ Γ∗, i ∈ I}

a

L

∶= {(xly, xry) ∣ x,y ∈ Γ∗, l

a

  • → r ∈ L}

gΣ(L) ∶= ⋃a∈Σ

L

a

❚♦ ✐♥❝r❡❛s❡ t❤❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r✱ ♦♥❡ ✉s❡s r❡❣✉❧❛r r❡str✐❝t✐♦♥ ♦♥ ✈❡rt✐❝❡s t♦ K ∈ Reg(Γ∗) ✿ gΣ(L, K) ∶= gΣ(L)∣K = gΣ(L) ∩ (K × Σ × K)✳ L ✜♥✐t❡ s✉❜❝❧❛s ♦❢ ❛✉t♦♠❛t✐❝ ❣r❛♣❤s L r❡❝♦❣♥✐s❛❜❧❡ s✉❜❝❧❛s ♦❢ r❛t✐♦♥❛❧ ❣r❛♣❤s

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✻ ✴ ✾✼

slide-101
SLIDE 101
  • r❛♣❤s ♦❢ ❧❛❜❡❧❧❡❞ ♣r❡✜①✲r❡✇r✐t✐♥❣ s②st❡♠s

L

a

→ ∶= {(ly, ry) ∣ y ∈ Γ∗, l

a

  • → r ∈ L}

pgΣ(L) ∶= ⋃a∈Σ

L

a

→ ♣r❡✜①✲r❡✇r✐t✐♥❣ s②st❡♠ ❣r❛♣❤ ✜♥✐t❡ ♣✉s❤❞♦✇♥✶ r❡❝♦❣♥✐s❛❜❧❡ ♣r❡✜①✕r❡❝♦❣♥✐s❛❜❧❡

✶✉♣ t♦ r❡str✐❝t✐♦♥ t♦ ❛ r❡❛❝❤❛❜❧❡ s✉❜❣r❛♣❤ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✼ ✴ ✾✼

slide-102
SLIDE 102

❍②♣❡r❡❞❣❡ r❡♣❧❛❝❡♠❡♥t s②st❡♠s

  • 1

A

  • 2
  • 1

A

  • 2 a

b

  • 3
  • 4
  • 3
  • 4

b

a

  • A
  • ω
  • a

a

  • b
  • b
  • b

b

  • a
  • a

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✽ ✴ ✾✼

slide-103
SLIDE 103

❊q✉❛t✐♦♥❛❧ ❣r❛♣❤s

X = ren p→2

q→4

⎛ ⎜ ⎝ add

2 a

  • →p

q a

  • →1

4 b

  • →q

p b

  • →3

(X + 1 + 3 ⊕ p ⊕ q) ⎞ ⎟ ⎠ ✇❤❡r❡ 1,2,3,4 ❛r❡ ♣♦rt ♥❛♠❡s G + p ♠❡❛♥s G ❡①t❡♥❞❡❞ ✇✐t❤ ♣♦rt p ✐❢ G ❤❛s ♥♦ ♣♦rt ♥❛♠❡❞ p✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✾ ✴ ✾✼

slide-104
SLIDE 104

❊①❡r❝✐s❡

X = ren q→p(add q b

  • →1(ren p→1(add p b
  • →q(X ⊕ q))))

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✵ ✴ ✾✼

slide-105
SLIDE 105

P❡tr✐ ♥❡t ❣r❛♣❤s

P❡tr✐ ♥❡t ⇔ ✈❡❝t♦r ❛❞❞✐t✐♦♥ s②st❡♠ ✭✈❛s✮ ❆ ✈❡❝t♦r ❛❞❞✐t✐♦♥ s②st❡♠ ✭♦❢ ❞✐♠❡♥s✐♦♥ n ∈ IN✮ ♦✈❡r Σ ✐s ✜♥✐t❡ s❡t ♦❢ ✈❡❝t♦rs V ⊂ Zn❛ t♦❣❡t❤❡r ✇✐t❤ ❛ ♠❛♣♣✐♥❣ V∶V → Σ . ❚❤❡ ❣r❛♣❤ ♦❢ ✈❛s V ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ str✉❝t✉r❡ A ♦✈❡r Σ ✿ A = INn A(a) = {(u,v) ∈ INn × INn ∣ v = u + t ∧ V(t) = a} ❢♦r a ∈ Σ

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✶ ✴ ✾✼

slide-106
SLIDE 106

❊①❛♠♣❧❡

n=2 Σ={a,b} V ={(1

1),(−2 −1)}

V((1

1)) = a

V((−2

−1)) = b (0,0) (0,1) (0,2) (0,3) (1,0) (1,1) (1,2) (1,3) a a a (2,0) (2,1) (2,2) (2,3) a a a b b b (3,0) (3,1) (3,2) (3,3) a a a b b b (4,0) (4,1) (4,2) (4,3) a a a b b b

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✷ ✴ ✾✼

slide-107
SLIDE 107

❈♦♠♣❛r✐s♦♥ ♦❢ ❣r❛♣❤ ❢❛♠✐❧❧❡s

✜♥✐t❡

  • ♣✉s❤❞♦✇♥
  • ❧✐♥❡❛r❧② ❜♦✉♥❞❡❞
  • P❡tr✐ ♥❡ts
  • r❡❣✉❧❛r
  • ❛✉t♦♠❛t✐❝
  • ♣r❡✜①✕r❡❝♦❣♥✐s❛❜❧❡
  • r❛t✐♦♥❛❧

r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡s

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✸ ✴ ✾✼

slide-108
SLIDE 108

❚r❛❝❡s

G =

  • 1

a

  • b
  • a
  • b
  • a
  • b
  • a
  • b
  • b
  • 2
  • c
  • c
  • c
  • c

❚r❛❝❡ ♦❢ G ❢r♦♠ {1} t♦ {2} ✿ L(G, {1}, {2}) = {anbcn ∣ n ∈ IN} ■♥ ❣❡♥❡r❛❧✱ ✇❡ ♠❛② ❝♦♥s✐❞❡r L(G, I, F) ✇❤❡r❡ I ❛♥❞ F ❜♦t❤ ▼❙❖✲❞❡✜♥❛❜❧❡ s❡ts ♦❢ ✈❡rt✐❝❡s ♦❢ G✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✹ ✴ ✾✼

slide-109
SLIDE 109

❣r❛♣❤ ❢❛♠✐❧✐❡s ❧❛♥❣❛❣❡s ❢❛♠✐❧❧❡s ✜♥✐t❡ r❡❣✉❧❛r ♣✉s❤❞♦✇♥ ❝♦♥t❡①t✲❢r❡❡ ❡q✉❛t✐♦♥❛❧ ❝♦♥t❡①t✲❢r❡❡ ♣r❡✜①✕r❡❝♦❣♥✐s❛❜❧❡ ❝♦♥t❡①t✲❢r❡❡ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ ❝♦♥t❡①t✲s❡♥s✐t✐✈❡ r❛t✐♦♥❛❧ ❝♦♥t❡①t✲s❡♥s✐t✐✈❡ r❡❝✉rs✐✈❡❧② ❡♥✉♠✳ r❡❝✉rs✐✈❡❧② ❡♥✉♠✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✺ ✴ ✾✼

slide-110
SLIDE 110

❈❛②❧❡②✲t②♣❡ ❣r❛♣❤s

str✐♥❣ r❡✇r✐t✐♥❣ s②st❡♠ S = {li → ri ∣ li,ri ∈ Γ∗, i ∈ I} s✐♥❣❧❡✲st❡♣ r❡✇r✐t✐♥❣ r❡❧❛t✐♦♥

S

∶= {(xly, xry) ∣ x,y ∈ Γ∗, l → r ∈ S} r❡✇r✐t✐♥❣ r❡❧❛t✐♦♥

S

❈❛②❧❡②✲t②♣❡ ❣r❛♣❤ CGΣ(S) ∶= {u

a

  • → v ∣ u,v ∈ Irr(S), a ∈ Σ, au

S v}

CGΣ(S, K) ∶= CGΣ(S)∣K

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✻ ✴ ✾✼

slide-111
SLIDE 111

❊①❛♠♣❧❡

S = { rR → Rr, rW → Wr, wR → Rw, wW → Ww, r✩ → R✩, w✩ → W✩, dR → ε, dW → ε } K = {R, W}∗✩ ✩

r

  • w

R✩

p

  • r
  • w

p

  • r
  • RR✩

p

  • r
  • w
  • RW✩

p r

  • w
  • WR✩

p

  • r
  • w

RRR✩

p

  • RRW✩

p

  • RWR✩

p

  • RWW✩

WRR✩

d

  • WR

p

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✼ ✴ ✾✼

slide-112
SLIDE 112

❈❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ ❣r❛♣❤s ❢❛♠✐❧✐❡s ✿ r❡str✐❝t✐♦♥s ♦♥ t❤❡ r❡✇r✐t✐♥❣

{CGΣ(S, K) ∣ S ✜♥✐t❡, K r❡❣✉❧❛r} = {r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡ ❣r❛♣❤s} ❲❡ ♥❡❡❞ t♦ ❛♥❛❧②③❡ t❤❡ r❡✇r✐t✐♥❣ ♦❢ ✇♦r❞s ♦❢ t❤❡ ❢♦r♠ au✱ ✇❤❡r❡ a ✐s ❛ ❧❡tt❡r ❛♥❞ u ∈ Irr(S) ✿

✶ r❡❞✉❝t✐♦♥✲❜♦✉♥❞❡❞✱ ✉♥✐t❛r② r❡❞✉❝t✐♦♥✲❜♦✉♥❞❡❞✱ ✷ ♣r❡✜①✲❜♦✉♥❞❡❞✱ ♣r❡✜①✱ ✸ ✐♥❝r❡❛s✐♥❣ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✽ ✴ ✾✼

slide-113
SLIDE 113

❝❧❛ss ♦❢ ❣r❛♣❤s ♣r❡s❡♥t❛t✐♦♥s ❝❛r❛❝t❡r✐③❛t✐♦♥ ❜② ❈❛②❧❡②✲t②♣❡ ❣r❛♣❤s CGΣ(S, K) ❞❡❝✐❞❛❜✐✲ ❧✐t② ♦❢ ▼❙❖ ❞❡❝✐❞❛❜✐✲ ❧✐t② ♦❢ ❋❖ ❝❧♦s✉r❡ ✉♥❞❡r s②♥❝❤r♦♥✐③❡❞ ♣r♦❞✉❝t ✜♥✐t❡ ❣r❛♣❤s ❝❧❛ss✐❝ K ∈ Fin❀ S ∈ Fin ❨❊❙ ❨❊❙ ❨❊❙ ♣✉s❤❞♦✇♥ ♣✉s❤❞♦✇♥ ❛✉t♦♠❛t❛ K ∈ Rat❀ S ∈ Fin ∩ 1RredB ❨❊❙ ❨❊❙ ◆❖ ♣r❡✜①✕ r❡❝♦❣♥✐s❛❜❧❡ ❣r❛♣❤s ♣r❡✜①✕ r❡❝♦❣♥✐s❛❜❧❡ r❡❧❛t✐♦♥s✱ s②st❡♠s ❱❘ ❡q✉❛t✐♦♥s K ∈ Rat❀ S ∈ Fin ∩ Pref ♦✉ S ∈ Rec ∩ PrefB ♦✉ . . . ❨❊❙ ❨❊❙ ◆❖ P❡tr✐ ♥❡t ❣r❛♣❤s P❡tr✐ ♥❡ts✱ ✈❡❝t♦r ❛❞❞✐t✐♦♥ s②st❡♠s ❄ ◆❖ ❨❊❙ ❨❊❙ ❛✉t♦♠❛t✐❝ ❣r❛♣❤s s②♥❝❤r♦♥✐③❡❞ tr❛♥s❞✉❝❡rs ❄ ◆❖ ❨❊❙ ❨❊❙ r❛t✐♦♥❛❧ ❣r❛♣❤s r❛t✐♦♥♥❛❧ tr❛♥s❞✉❝❡rs K ∈ Rat❀ S ∈ Fin ∩ LInc ♦✉ S ∈ Rec ∩ LInc ♦✉ . . . ◆❖ ◆❖ ❨❊❙ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ ❣r❛♣❤s ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ ❛✉t♦♠❛t❛ K ∈ Rat❀ S ∈ Fin ∩ LB ◆❖ ◆❖ ❨❊❙ r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡s ❣r❛♣❤s ❚✉r✐♥❣ ♠❛❝❤✐♥❡s K ∈ Rat❀ S ∈ Fin ◆❖ ◆❖ ❨❊❙ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽✾ ✴ ✾✼

slide-114
SLIDE 114

❚r❡❡s

❚❤❡♦r❡♠ ✭❈❛✉❝❛❧ ✾✻✮

❊✈❡r② ♣r❡✜①✕r❡❝♦❣♥✐s❛❜❧❡ ❣r❛♣❤ ✐s ▼❙❖✲✐♥t❡r♣r❡t❛❜❧❡ ✇✐t❤✐♥ t❤❡ ❝♦♠♣❧❡t❡ ✐♥✜♥✐t❡ ❜✐♥❛r② tr❡❡✳

  • 1
  • 1
  • 1
  • ❚❤❡♦r❡♠ ✭❇❛rt❤❡❧♠❛♥♥ ✾✽✱ ❇❧✉♠❡♥s❛t❤ ✵✶✮

❊✈❡r② ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❝♦♠♣❧❡t❡ ✐♥✜♥✐t❡ ❜✐♥❛r② tr❡❡ ❜② ❛♥ ▼❙❖✲tr❛♥s❞✉❝t✐♦♥ ✐s ♣r❡✜①✕r❡❝♦❣♥✐s❛❜❧❡✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾✵ ✴ ✾✼

slide-115
SLIDE 115

❍♦✇ ❝❛♥ ♦♥❡ ♦❜t❛✐♥ ♠♦r❡ ❣❡♥❡r❛❧ ❣r❛♣❤ ❢❛♠✐❧✐❡s ✇✐t❤ ❞❡❝✐❞❛❜❧❡ ▼❙❖ t❤❡♦r②❄

▼❙❖✕❝♦♠♣❛t✐❜❧❡ str✉❝t✉r❡✲❜✉✐❧❞✐♥❣ ♦♣❡r❛t✐♦♥s

✉♥❢♦❧❞✐♥❣ ❬❈♦✉r❝❡❧❧❡ ✫ ❲❛❧✉❦✐❡✇✐❝③ ✾✺❪ ✐t❡r❛t✐♦♥ ❬▼✉❝❤♥✐❦ ✽✹❪✱ ❬❲❛❧✉❦✐❡✇✐❝③ ✾✻❪

✭♣r♦❣r❛♠✮ s❝❤❡♠❡s✴tr❡❡ ❣r❛♠♠❛rs❄

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾✶ ✴ ✾✼

slide-116
SLIDE 116

▲❡✈❡❧✲✵ s❝❤❡♠❡

♦♥❧② ♦♥❡ ❜❛s❡ t②♣❡ 0 s✐❣♥❛t✉r❡ = f ∶0 → 0 → 0 g ∶0 → 0 → 0 a ∶0 b ∶0 ♣r♦❝❡❞✉r❡s = F ∶0 G∶0 F = fGb G = gaF F = f G b G = g a F ❚❤❡ ❧❡❛st s♦❧✉t✐♦♥ ✐♥ F f g b a f g b a f

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾✷ ✴ ✾✼

slide-117
SLIDE 117

▲❡✈❡❧✲✶ s❝❤❡♠❡

Pr♦❝❡❞✉r❡s ♠❛② ❝❛rr② ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❜❛s❡ t②♣❡✳ S = Fa Fx = f(F(hx))x = f F x h x s✐❣♥❛t✉r❡✿ f ∶0 → 0 → 0, h∶0 → 0, a∶0 ♣r♦❝❡❞✉r❡s✿ S ∶0, F ∶0 → 0 ♣❛r❛♠❡t❡r✿ x∶0 t❤❡ ❧❡❛st s♦✲ ❧✉t✐♦♥ ✐♥ S f f a f h h a h a

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾✸ ✴ ✾✼

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SLIDE 118

▲❡✈❡❧✲✷ s❝❤❡♠❡

P❛r❛♠❡t❡rs ♠❛② ❤❛✈❡ ❢✉♥❝t✐♦♥❛❧ t②♣❡s✳ S = Fha Fϕx = f(ϕ(F(λz.ϕ(hz))x))x s✐❣♥❛t✉r❡✿ f ∶0 → 0 → 0, h∶0 → 0, a∶0 ♣r♦❝❡❞✉r❡s✿ S ∶0, F ∶(0 → 0) → 0 → 0 ♣❛r❛♠❡t❡rs✿ ϕ∶0 → 0, x∶0 f h a f h a h f a

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾✹ ✴ ✾✼

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SLIDE 119

❉❡❝✐❞❛❜✐❧✐t② ♦❢ ▼❙❖ t❤❡♦r②

  • ▲❡✈❡❧✲✵ s❝❤❡♠❡s✱

r❡❣✉❧❛r tr❡❡s ❬❘❛❜✐♥ ✻✾❪

  • ▲❡✈❡❧✲✶ s❝❤❡♠❡s✱

❛❧❣❡❜r❛✐❝ tr❡❡s ❬❈♦✉r❝❡❧❧❡ ✾✺❪

  • ▲❡✈❡❧✲✷ s❛❢❡ s❝❤❡♠❡s✱

s❛❢❡ ❤②♣❡r❛❧❣❡❜r❛✐❝ tr❡❡s ❬❑✳✱ ◆✐✇✐➠s❦✐✱ ❯r③②❝③②♥ ✵✶❪

  • ▲❡✈❡❧✲n s❛❢❡ s❝❤❡♠❡s✱

s❛❢❡ ❤②♣❡r❛❧❣❡❜r❛✐❝ tr❡❡s ❬❑✳✱ ◆✐✇✐➠s❦✐✱ ❯r③②❝③②♥ ✵✷❪

  • ▲❡✈❡❧✲n s❝❤❡♠❡s✱

❤②♣❡r❛❧❣❡❜r❛✐❝ tr❡❡s ❬❆❡❝❤❧✐❣✱ ❖♥❣ ✵✺❪

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾✺ ✴ ✾✼

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SLIDE 120

❙❛❢❡ s❝❤❡♠❡s

❆ s❝❤❡♠❡ ✐s ✉♥s❛❢❡ ✐❢ ❛♥ ❛r❣✉♠❡♥t t ♦❢ ❛ ♣r♦❝❡❞✉r❡ ❝❛❧❧ ❝♦♥t❛✐♥s ❛ ❣❧♦❜❛❧ ♣❛r❛♠❡t❡r ♦❢ ❧❡✈❡❧ str✐❝t❧② ❧♦✇❡r t❤❛♥ t✳ ❊①❛♠♣❧❡ ♦❢ ❛♥ ✉♥s❛❢❡ s❝❤❡♠❡✿ S =F(h,a) F(ϕ,x) =f(F(λz.f(ϕ(x),z),x),ϕ(x)) s✐❣♥❛t✉r❡ ✿ f ∶0 → 0 → 0, h∶0 → 0, a∶0 ♣r♦❝❡❞✉r❡♥❛♠❡s ✿ S ∶0, F ∶(0 → 0) → 0 → 0 ♣❛r❛♠❡t❡rs ✿ ϕ∶0 → 0, x∶0

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾✻ ✴ ✾✼

slide-121
SLIDE 121

❊q✉✐✈❛❧❡♥❝❡ ✇✐t❤ ❤✐❣❤❡r✲♦r❞❡r ♣✉s❤❞♦✇♥ tr❡❡ ❛✉t♦♠❛t❛

❧❡✈❡❧✲n s❛❢❡ s❝❤❡♠❡s ⇔ ❧❡✈❡❧✲n ♣✉s❤❞♦✇♥ tr❡❡ ❛✉t♦♠❛t❛ ❧❡✈❡❧✲n s❝❤❡♠❡s ⇔ ❧❡✈❡❧✲n ♣✉s❤❞♦✇♥ tr❡❡ ❛✉t♦♠❛t❛ ✇✐t❤ ♣❛♥✐❝ ⇔ ❑r✐✈✐♥❡✬s ♠❛❝❤✐♥❡s ❆✉t♦♠❛ ✇✐t❤ ♣❛♥✐❝ ❛r❡ s♦♠❡t✐♠❡s ❝❛❧❧❡❞ ❝♦❧❧❛♣s✐❜❧❡ ❛✉t♦♠❛t❛✳

❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾✼ ✴ ✾✼