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Oct 16, 2022 209 likes •1.2k views

s t ss s rs r rst rs

slide-1
SLIDE 1

■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ❉▲❚ ✷✵✶✷

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-2
SLIDE 2

❊①t❡♥❞❡❞ ❘❡❣✉❧❛r ❊①♣r❡ss✐♦♥s ✇✐t❤ ❇❛❝❦r❡❢❡r❡♥❝❡s ✭❘❊●❊❳✮

❘❊●❊❳ ❛r❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳

✶ ❛

❜

✶❝ ❜

✶ ✇❝❜♥✇ ✇ ❛ ❜ ♥ ✵

✶ ✷ ❛ ✷ ❜ ✷ ✶ ❝ ✶

❛♥❜❛♥❝❛♥❜❛♥ ♥ ✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-3
SLIDE 3

❊①t❡♥❞❡❞ ❘❡❣✉❧❛r ❊①♣r❡ss✐♦♥s ✇✐t❤ ❇❛❝❦r❡❢❡r❡♥❝❡s ✭❘❊●❊❳✮

❘❊●❊❳ ❛r❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳ (✶ (❛ | ❜)∗ )✶❝ ❜∗\✶ ✇❝❜♥✇ ✇ ❛ ❜ ♥ ✵

✶ ✷ ❛ ✷ ❜ ✷ ✶ ❝ ✶

❛♥❜❛♥❝❛♥❜❛♥ ♥ ✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-4
SLIDE 4

❊①t❡♥❞❡❞ ❘❡❣✉❧❛r ❊①♣r❡ss✐♦♥s ✇✐t❤ ❇❛❝❦r❡❢❡r❡♥❝❡s ✭❘❊●❊❳✮

❘❊●❊❳ ❛r❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳ (✶ (❛ | ❜)∗ )✶❝ ❜∗\✶ {✇❝❜♥✇ | ✇ ∈ {❛, ❜}∗, ♥ ≥ ✵}

✶ ✷ ❛ ✷ ❜ ✷ ✶ ❝ ✶

❛♥❜❛♥❝❛♥❜❛♥ ♥ ✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-5
SLIDE 5

❊①t❡♥❞❡❞ ❘❡❣✉❧❛r ❊①♣r❡ss✐♦♥s ✇✐t❤ ❇❛❝❦r❡❢❡r❡♥❝❡s ✭❘❊●❊❳✮

❘❊●❊❳ ❛r❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳ (✶ (❛ | ❜)∗ )✶❝ ❜∗\✶ {✇❝❜♥✇ | ✇ ∈ {❛, ❜}∗, ♥ ≥ ✵}

✶ ✷ ❛ ✷ ❜ ✷ ✶ ❝ ✶

❛♥❜❛♥❝❛♥❜❛♥ ♥ ✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-6
SLIDE 6

❊①t❡♥❞❡❞ ❘❡❣✉❧❛r ❊①♣r❡ss✐♦♥s ✇✐t❤ ❇❛❝❦r❡❢❡r❡♥❝❡s ✭❘❊●❊❳✮

❘❊●❊❳ ❛r❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳ (✶ (❛ | ❜)∗ )✶❝ ❜∗\✶ {✇❝❜♥✇ | ✇ ∈ {❛, ❜}∗, ♥ ≥ ✵}

✶ ✷ ❛ ✷ ❜ ✷ ✶ ❝ ✶

❛♥❜❛♥❝❛♥❜❛♥ ♥ ✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-7
SLIDE 7

❊①t❡♥❞❡❞ ❘❡❣✉❧❛r ❊①♣r❡ss✐♦♥s ✇✐t❤ ❇❛❝❦r❡❢❡r❡♥❝❡s ✭❘❊●❊❳✮

❘❊●❊❳ ❛r❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳ (✶ (❛ | ❜)∗ )✶❝ ❜∗\✶ {✇❝❜♥✇ | ✇ ∈ {❛, ❜}∗, ♥ ≥ ✵}

✶ ✷ ❛ ✷ ❜ ✷ ✶ ❝ ✶

❛♥❜❛♥❝❛♥❜❛♥ ♥ ✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-8
SLIDE 8

❊①t❡♥❞❡❞ ❘❡❣✉❧❛r ❊①♣r❡ss✐♦♥s ✇✐t❤ ❇❛❝❦r❡❢❡r❡♥❝❡s ✭❘❊●❊❳✮

❘❊●❊❳ ❛r❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳ (✶ (❛ | ❜)∗ )✶❝ ❜∗\✶ {✇❝❜♥✇ | ✇ ∈ {❛, ❜}∗, ♥ ≥ ✵} (✶ (✷ ❛∗ )✷ ❜ \✷ )✶ ❝\✶ {❛♥❜❛♥❝❛♥❜❛♥ | ♥ ≥ ✵}

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-9
SLIDE 9

Pr❛❝t✐❝❛❧ ❘❡❧❡✈❛♥❝❡ ♦❢ ❘❊●❊❳

❘❊●❊❳ ❛r❡ ✐♥t❡♥s❡❧② ❛♣♣❧✐❡❞ ✐♥ ♣r❛❝t✐❝❡✳✳✳

❚r❛❞✐t✐♦♥❛❧ ❛♥❞ ▼♦❞❡r♥ ❣r❡♣ ✈✐ ▼♦❞❡r♥ s❡❞

  • ◆❯ ❊♠❛❝s

P❡r❧ P②t❤♦♥ ❏❛✈❛ .◆❡t

✳✳✳❡✈❡♥ t❤♦✉❣❤ t❤❡✐r ♠❡♠❜❡rs❤✐♣ ♣r♦❜❧❡♠ ✐s ◆P✲❝♦♠♣❧❡t❡✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-10
SLIDE 10

Pr❛❝t✐❝❛❧ ❘❡❧❡✈❛♥❝❡ ♦❢ ❘❊●❊❳

❘❊●❊❳ ❛r❡ ✐♥t❡♥s❡❧② ❛♣♣❧✐❡❞ ✐♥ ♣r❛❝t✐❝❡✳✳✳

❚r❛❞✐t✐♦♥❛❧ ❛♥❞ ▼♦❞❡r♥ ❣r❡♣ ✈✐ ▼♦❞❡r♥ s❡❞

  • ◆❯ ❊♠❛❝s

P❡r❧ P②t❤♦♥ ❏❛✈❛ .◆❡t

✳✳✳❡✈❡♥ t❤♦✉❣❤ t❤❡✐r ♠❡♠❜❡rs❤✐♣ ♣r♦❜❧❡♠ ✐s ◆P✲❝♦♠♣❧❡t❡✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-11
SLIDE 11

❆♥ ❊❛s② ❊①❛♠♣❧❡

(✶ (❛ | ❜)∗❝∗ )✶ (✷ (❜ | ❞) )✷ (\✶ | \✷) () ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-12
SLIDE 12

❆♥ ❊❛s② ❊①❛♠♣❧❡

(✶ (❛ | ❜)∗❝∗ )✶ (✷ (❜ | ❞) )✷ ( \✶ | \✷ ) () ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-13
SLIDE 13

❆♥ ❊❛s② ❊①❛♠♣❧❡

(✶ (❛ | ❜)∗❝∗ )✶ (✷ (❜ | ❞) )✷ ( \✶ | \✷ ) ①✶ ①✷ ( ①✶ | ①✷ ) () ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-14
SLIDE 14

❆♥ ❊❛s② ❊①❛♠♣❧❡

(✶ (❛ | ❜)∗❝∗ )✶ (✷ (❜ | ❞) )✷ ( \✶ | \✷ ) ①✶ ①✷ ( ①✶ | ①✷ ) L((❛ | ❜)∗❝∗) L(❜ | ❞) () ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-15
SLIDE 15

❆ ▼♦r❡ ❈♦♠♣❧❡① ❊①❛♠♣❧❡

(✶ (✷ (✸ (❛ | ❜)∗ )✸ ❝\✸ )✷ ❞ ( \✷ ❝)∗ )✶ ❡ \✶ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-16
SLIDE 16

❆ ▼♦r❡ ❈♦♠♣❧❡① ❊①❛♠♣❧❡

(✶ (✷ (✸ (❛ | ❜)∗ )✸ ❝\✸ )✷ ❞ ( \✷ ❝)∗ )✶ ❡ \✶ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-17
SLIDE 17

❆ ▼♦r❡ ❈♦♠♣❧❡① ❊①❛♠♣❧❡

(✶ (✷ (✸ (❛ | ❜)∗ )✸ ❝\✸ )✷ ❞ ( \✷ ❝)∗ )✶ ❡ \✶ ①✶ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-18
SLIDE 18

❆ ▼♦r❡ ❈♦♠♣❧❡① ❊①❛♠♣❧❡

(✶ (✷ (✸ (❛ | ❜)∗ )✸ ❝\✸ )✷ ❞ ( \✷ ❝)∗ )✶ ❡ \✶ ①✶ ①✷❞(①✷❝)∗ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 19

❆ ▼♦r❡ ❈♦♠♣❧❡① ❊①❛♠♣❧❡

(✶ (✷ (✸ (❛ | ❜)∗ )✸ ❝\✸ )✷ ❞ ( \✷ ❝)∗ )✶ ❡ \✶ ①✶ ①✷❞(①✷❝)∗ ①✸ ❝ ①✸ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 20

❆ ▼♦r❡ ❈♦♠♣❧❡① ❊①❛♠♣❧❡

(✶ (✷ (✸ (❛ | ❜)∗ )✸ ❝\✸ )✷ ❞ ( \✷ ❝)∗ )✶ ❡ \✶ ①✶ ①✷❞(①✷❝)∗ ①✸ ❝ ①✸ L((❛ | ❜)∗)

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 21

❆ ❋❛✐r❧② ■♥✈♦❧✈❡❞ ❊①❛♠♣❧❡

(✶ ❛∗ )✶ ❝ (✷ \✶ (❜ | ❝) \✶ )✷ ❝ (✸ (\✶ | ❞)∗ \✷ )✸ \✸ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 22

❆ ❋❛✐r❧② ■♥✈♦❧✈❡❞ ❊①❛♠♣❧❡

(✶ ❛∗ )✶ ❝ (✷ \✶ (❜ | ❝) \✶ )✷ ❝ (✸ (\✶ | ❞)∗ \✷ )✸ \✸ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 23

❆ ❋❛✐r❧② ■♥✈♦❧✈❡❞ ❊①❛♠♣❧❡

(✶ ❛∗ )✶ ❝ (✷ \✶ (❜ | ❝) \✶ )✷ ❝ (✸ (\✶ | ❞)∗ \✷ )✸ \✸ ①✶ ①✷ ①✸ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 24

❆ ❋❛✐r❧② ■♥✈♦❧✈❡❞ ❊①❛♠♣❧❡

(✶ ❛∗ )✶ ❝ (✷ \✶ (❜ | ❝) \✶ )✷ ❝ (✸ (\✶ | ❞)∗ \✷ )✸ \✸ ①✶ ①✷ ①✸ L(❛∗) ①✶(❜ | ❝)①✶ (①✶| ❞)∗ ①✷ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 25

❆ ❋❛✐r❧② ■♥✈♦❧✈❡❞ ❊①❛♠♣❧❡

(✶ ❛∗ )✶ ❝ (✷ \✶ (❜ | ❝) \✶ )✷ ❝ (✸ (\✶ | ❞)∗ \✷ )✸ \✸ ①✶ ①✷ ①✸ L(❛∗) ①✶(❜ | ❝)①✶ (①✶| ❞)∗ ①✷ L(❛∗) L(❛∗) ①✶(❜ | ❝)①✶ ()

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 26

❆ ❋❛✐r❧② ■♥✈♦❧✈❡❞ ❊①❛♠♣❧❡

(✶ ❛∗ )✶ ❝ (✷ \✶ (❜ | ❝) \✶ )✷ ❝ (✸ (\✶ | ❞)∗ \✷ )✸ \✸ ①✶ ①✷ ①✸ L(❛∗) ①✶(❜ | ❝)①✶ (①✶| ❞)∗ ①✷ L(❛∗) L(❛∗) ①✶(❜ | ❝)①✶ L(❛∗)

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 27

❆✐♠ ♦❢ ❚❤✐s P❛♣❡r

❘❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ♦♥ t❤❡ ♦♥❡ ❤❛♥❞ ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t ♦♥ t❤❡ ♦t❤❡r ❛ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ❝♦♥❝❡♣ts ✐♥ ❧❛♥❣✉❛❣❡ t❤❡♦r②✳ ■♥ ❘❊●❊❳✱ t❤❡s❡ t✇♦ ❝♦♥❝❡♣ts s❡❡♠ ✐♥❤❡r❡♥t❧② ❡♥t❛♥❣❧❡❞ ❛♥❞ ✐t s❡❡♠s ❞✐✣❝✉❧t② t♦ tr❡❛t t❤❡♠ s❡♣❛r❛t❡❧②✳ ❖✉r ❛♣♣r♦❛❝❤✿ ❙t✉❞② ❘❊●❊❳ ❜② ✐♥✈❡st✐❣❛t✐♥❣ ❛❧t❡r♥❛t✐✈❡ ✇❛②s t♦ ❝♦♠❜✐♥❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳✳✳ ✳✳✳✇✐t❤♦✉t ❡①❝❡❡❞✐♥❣ t❤❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s✳ ■♥❢♦r♠❛❧❧②✿ ❚❛❦❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✱ t❛❦❡ s♦♠❡ ♠❡❝❤❛♥✐s♠ ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✱ ❝♦♠❜✐♥❡ t❤❡♠ ❛♥❞ s❡❡ ❤♦✇ ♠✉❝❤ ♦❢ t❤❡ ❝❧❛ss ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s ✇❡ ❛❝t✉❛❧❧② ❣❡t✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 28

❆✐♠ ♦❢ ❚❤✐s P❛♣❡r

❘❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ♦♥ t❤❡ ♦♥❡ ❤❛♥❞ ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t ♦♥ t❤❡ ♦t❤❡r ❛ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ❝♦♥❝❡♣ts ✐♥ ❧❛♥❣✉❛❣❡ t❤❡♦r②✳ ■♥ ❘❊●❊❳✱ t❤❡s❡ t✇♦ ❝♦♥❝❡♣ts s❡❡♠ ✐♥❤❡r❡♥t❧② ❡♥t❛♥❣❧❡❞ ❛♥❞ ✐t s❡❡♠s ❞✐✣❝✉❧t② t♦ tr❡❛t t❤❡♠ s❡♣❛r❛t❡❧②✳ ❖✉r ❛♣♣r♦❛❝❤✿ ❙t✉❞② ❘❊●❊❳ ❜② ✐♥✈❡st✐❣❛t✐♥❣ ❛❧t❡r♥❛t✐✈❡ ✇❛②s t♦ ❝♦♠❜✐♥❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳✳✳ ✳✳✳✇✐t❤♦✉t ❡①❝❡❡❞✐♥❣ t❤❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s✳ ■♥❢♦r♠❛❧❧②✿ ❚❛❦❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✱ t❛❦❡ s♦♠❡ ♠❡❝❤❛♥✐s♠ ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✱ ❝♦♠❜✐♥❡ t❤❡♠ ❛♥❞ s❡❡ ❤♦✇ ♠✉❝❤ ♦❢ t❤❡ ❝❧❛ss ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s ✇❡ ❛❝t✉❛❧❧② ❣❡t✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 29

❆✐♠ ♦❢ ❚❤✐s P❛♣❡r

❘❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ♦♥ t❤❡ ♦♥❡ ❤❛♥❞ ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t ♦♥ t❤❡ ♦t❤❡r ❛ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ❝♦♥❝❡♣ts ✐♥ ❧❛♥❣✉❛❣❡ t❤❡♦r②✳ ■♥ ❘❊●❊❳✱ t❤❡s❡ t✇♦ ❝♦♥❝❡♣ts s❡❡♠ ✐♥❤❡r❡♥t❧② ❡♥t❛♥❣❧❡❞ ❛♥❞ ✐t s❡❡♠s ❞✐✣❝✉❧t② t♦ tr❡❛t t❤❡♠ s❡♣❛r❛t❡❧②✳ ❖✉r ❛♣♣r♦❛❝❤✿ ❙t✉❞② ❘❊●❊❳ ❜② ✐♥✈❡st✐❣❛t✐♥❣ ❛❧t❡r♥❛t✐✈❡ ✇❛②s t♦ ❝♦♠❜✐♥❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳✳✳ ✳✳✳✇✐t❤♦✉t ❡①❝❡❡❞✐♥❣ t❤❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s✳ ■♥❢♦r♠❛❧❧②✿ ❚❛❦❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✱ t❛❦❡ s♦♠❡ ♠❡❝❤❛♥✐s♠ ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✱ ❝♦♠❜✐♥❡ t❤❡♠ ❛♥❞ s❡❡ ❤♦✇ ♠✉❝❤ ♦❢ t❤❡ ❝❧❛ss ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s ✇❡ ❛❝t✉❛❧❧② ❣❡t✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 30

❆✐♠ ♦❢ ❚❤✐s P❛♣❡r

❘❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ♦♥ t❤❡ ♦♥❡ ❤❛♥❞ ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t ♦♥ t❤❡ ♦t❤❡r ❛ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ❝♦♥❝❡♣ts ✐♥ ❧❛♥❣✉❛❣❡ t❤❡♦r②✳ ■♥ ❘❊●❊❳✱ t❤❡s❡ t✇♦ ❝♦♥❝❡♣ts s❡❡♠ ✐♥❤❡r❡♥t❧② ❡♥t❛♥❣❧❡❞ ❛♥❞ ✐t s❡❡♠s ❞✐✣❝✉❧t② t♦ tr❡❛t t❤❡♠ s❡♣❛r❛t❡❧②✳ ❖✉r ❛♣♣r♦❛❝❤✿ ❙t✉❞② ❘❊●❊❳ ❜② ✐♥✈❡st✐❣❛t✐♥❣ ❛❧t❡r♥❛t✐✈❡ ✇❛②s t♦ ❝♦♠❜✐♥❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳✳✳ ✳✳✳✇✐t❤♦✉t ❡①❝❡❡❞✐♥❣ t❤❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s✳ ■♥❢♦r♠❛❧❧②✿ ❚❛❦❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✱ t❛❦❡ s♦♠❡ ♠❡❝❤❛♥✐s♠ ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✱ ❝♦♠❜✐♥❡ t❤❡♠ ❛♥❞ s❡❡ ❤♦✇ ♠✉❝❤ ♦❢ t❤❡ ❝❧❛ss ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s ✇❡ ❛❝t✉❛❧❧② ❣❡t✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-31
SLIDE 31

❆✐♠ ♦❢ ❚❤✐s P❛♣❡r

❘❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ♦♥ t❤❡ ♦♥❡ ❤❛♥❞ ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t ♦♥ t❤❡ ♦t❤❡r ❛ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ❝♦♥❝❡♣ts ✐♥ ❧❛♥❣✉❛❣❡ t❤❡♦r②✳ ■♥ ❘❊●❊❳✱ t❤❡s❡ t✇♦ ❝♦♥❝❡♣ts s❡❡♠ ✐♥❤❡r❡♥t❧② ❡♥t❛♥❣❧❡❞ ❛♥❞ ✐t s❡❡♠s ❞✐✣❝✉❧t② t♦ tr❡❛t t❤❡♠ s❡♣❛r❛t❡❧②✳ ❖✉r ❛♣♣r♦❛❝❤✿ ❙t✉❞② ❘❊●❊❳ ❜② ✐♥✈❡st✐❣❛t✐♥❣ ❛❧t❡r♥❛t✐✈❡ ✇❛②s t♦ ❝♦♠❜✐♥❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✳✳✳ ✳✳✳✇✐t❤♦✉t ❡①❝❡❡❞✐♥❣ t❤❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s✳ ■♥❢♦r♠❛❧❧②✿ ❚❛❦❡ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✱ t❛❦❡ s♦♠❡ ♠❡❝❤❛♥✐s♠ ♦❢ ❤♦♠♦♠♦r♣❤✐❝ r❡♣❧❛❝❡♠❡♥t✱ ❝♦♠❜✐♥❡ t❤❡♠ ❛♥❞ s❡❡ ❤♦✇ ♠✉❝❤ ♦❢ t❤❡ ❝❧❛ss ♦❢ ❘❊●❊❳ ❧❛♥❣✉❛❣❡s ✇❡ ❛❝t✉❛❧❧② ❣❡t✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 32

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

P❛tt❡r♥✿ ❆ ✇♦r❞ ❝♦♥t❛✐♥✐♥❣ t❡r♠✐♥❛❧s ✭❡✳ ❣✳ Σ = {❛, ❜, ❝}✮ ❛♥❞ ✈❛r✐❛❜❧❡s ✭❳ := {①✶, ①✷, ①✸, . . .}✮✳

①✶ ①✷ ❜ ①✶ ①✷ ①✶ ✇ ✇ ✉ ✈ ❜ ✉ ✈ ✉ ✉ ✈ ✳ ❆ t②♣❡ ❢♦r ✿ ❚①✶ ❚①✷ ✇ ✇ ✉ ✈ ❜ ✉ ✈ ✉ ✉ ❚①✶ ✈ ❚①✷ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 33

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

P❛tt❡r♥✿ ❆ ✇♦r❞ ❝♦♥t❛✐♥✐♥❣ t❡r♠✐♥❛❧s ✭❡✳ ❣✳ Σ = {❛, ❜, ❝}✮ ❛♥❞ ✈❛r✐❛❜❧❡s ✭❳ := {①✶, ①✷, ①✸, . . .}✮✳

α = ①✶ ①✷ ❜ ①✶ ①✷ ①✶ ✇ ✇ ✉ ✈ ❜ ✉ ✈ ✉ ✉ ✈ ✳ ❆ t②♣❡ ❢♦r ✿ ❚①✶ ❚①✷ ✇ ✇ ✉ ✈ ❜ ✉ ✈ ✉ ✉ ❚①✶ ✈ ❚①✷ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 34

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

P❛tt❡r♥✿ ❆ ✇♦r❞ ❝♦♥t❛✐♥✐♥❣ t❡r♠✐♥❛❧s ✭❡✳ ❣✳ Σ = {❛, ❜, ❝}✮ ❛♥❞ ✈❛r✐❛❜❧❡s ✭❳ := {①✶, ①✷, ①✸, . . .}✮✳

α = ①✶ ①✷ ❜ ①✶ ①✷ ①✶ LΣ(α) = {✇ | ✇ = ✉ ✈ ❜ ✉ ✈ ✉, ✉, ✈ ∈ Σ∗}✳ ❆ t②♣❡ ❢♦r ✿ ❚①✶ ❚①✷ ✇ ✇ ✉ ✈ ❜ ✉ ✈ ✉ ✉ ❚①✶ ✈ ❚①✷ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 35

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

P❛tt❡r♥✿ ❆ ✇♦r❞ ❝♦♥t❛✐♥✐♥❣ t❡r♠✐♥❛❧s ✭❡✳ ❣✳ Σ = {❛, ❜, ❝}✮ ❛♥❞ ✈❛r✐❛❜❧❡s ✭❳ := {①✶, ①✷, ①✸, . . .}✮✳

α = ①✶ ①✷ ❜ ①✶ ①✷ ①✶ LΣ(α) = {✇ | ✇ = ✉ ✈ ❜ ✉ ✈ ✉, ✉, ✈ ∈ Σ∗}✳ ❆ t②♣❡ ❢♦r α✿ T := (❚①✶, ❚①✷) ✇ ✇ ✉ ✈ ❜ ✉ ✈ ✉ ✉ ❚①✶ ✈ ❚①✷ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 36

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

P❛tt❡r♥✿ ❆ ✇♦r❞ ❝♦♥t❛✐♥✐♥❣ t❡r♠✐♥❛❧s ✭❡✳ ❣✳ Σ = {❛, ❜, ❝}✮ ❛♥❞ ✈❛r✐❛❜❧❡s ✭❳ := {①✶, ①✷, ①✸, . . .}✮✳

α = ①✶ ①✷ ❜ ①✶ ①✷ ①✶ LΣ(α) = {✇ | ✇ = ✉ ✈ ❜ ✉ ✈ ✉, ✉, ✈ ∈ Σ∗}✳ ❆ t②♣❡ ❢♦r α✿ T := (❚①✶, ❚①✷) LT (α) = {✇ | ✇ = ✉ ✈ ❜ ✉ ✈ ✉, ✉ ∈ ❚①✶, ✈ ∈ ❚①✷}✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 37

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

P❆❚ := (Σ ∪ ❳)+✳ ✈❛r ✿ ❙❡t ♦❢ ✈❛r✐❛❜❧❡s ♦❝❝✉rr✐♥❣ ✐♥ ✳ ❊✳ ❣✳ ✈❛r ①✶❛❜①✷❜❛①✶①✷❝①✸ ①✶ ①✷ ①✸ ✳ ❋♦r ❛♥② ❧❛♥❣✉❛❣❡ ❝❧❛ss ✱ P❆❚ P❆❚

✈❛r

✳ Pr♦♣♦s✐t✐♦♥

❘❊● P❆❚

❘❊●❊❳ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 38

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

P❆❚ := (Σ ∪ ❳)+✳ ✈❛r(α)✿ ❙❡t ♦❢ ✈❛r✐❛❜❧❡s ♦❝❝✉rr✐♥❣ ✐♥ α✳ ❊✳ ❣✳ ✈❛r(①✶❛❜①✷❜❛①✶①✷❝①✸) = {①✶, ①✷, ①✸}✳ ❋♦r ❛♥② ❧❛♥❣✉❛❣❡ ❝❧❛ss ✱ P❆❚ P❆❚

✈❛r

✳ Pr♦♣♦s✐t✐♦♥

❘❊● P❆❚

❘❊●❊❳ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 39

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

P❆❚ := (Σ ∪ ❳)+✳ ✈❛r(α)✿ ❙❡t ♦❢ ✈❛r✐❛❜❧❡s ♦❝❝✉rr✐♥❣ ✐♥ α✳ ❊✳ ❣✳ ✈❛r(①✶❛❜①✷❜❛①✶①✷❝①✸) = {①✶, ①✷, ①✸}✳ ❋♦r ❛♥② ❧❛♥❣✉❛❣❡ ❝❧❛ss L✱ LL(P❆❚) := {LT (α) | α ∈ P❆❚, T ∈ L| ✈❛r(α)|}✳ Pr♦♣♦s✐t✐♦♥

❘❊● P❆❚

❘❊●❊❳ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 40

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

P❆❚ := (Σ ∪ ❳)+✳ ✈❛r(α)✿ ❙❡t ♦❢ ✈❛r✐❛❜❧❡s ♦❝❝✉rr✐♥❣ ✐♥ α✳ ❊✳ ❣✳ ✈❛r(①✶❛❜①✷❜❛①✶①✷❝①✸) = {①✶, ①✷, ①✸}✳ ❋♦r ❛♥② ❧❛♥❣✉❛❣❡ ❝❧❛ss L✱ LL(P❆❚) := {LT (α) | α ∈ P❆❚, T ∈ L| ✈❛r(α)|}✳ Pr♦♣♦s✐t✐♦♥ L❘❊●(P❆❚) ⊆ L(❘❊●❊❳)✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 41

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

■❞❡❛✿ L✶ := L❘❊●(P❆❚)✱

✷

✶ P❆❚ ✱

✸

✷ P❆❚ ✱

✳ ✳ ✳ Pr♦♣♦s✐t✐♦♥ ❋♦r ❛♥② ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s ✱ P❆❚

P❆❚ P❆❚ ✳

❍❡♥❝❡✱ t❤❡ ❛s♣❡❝t ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❝❛♥♥♦t ❜❡ ❧✐♠✐t❡❞ t♦ t❤❡ t②♣❡ ❧❛♥❣✉❛❣❡s✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-42
SLIDE 42

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

■❞❡❛✿ L✶ := L❘❊●(P❆❚)✱ L✷ := LL✶(P❆❚)✱

✸

✷ P❆❚ ✱

✳ ✳ ✳ Pr♦♣♦s✐t✐♦♥ ❋♦r ❛♥② ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s ✱ P❆❚

P❆❚ P❆❚ ✳

❍❡♥❝❡✱ t❤❡ ❛s♣❡❝t ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❝❛♥♥♦t ❜❡ ❧✐♠✐t❡❞ t♦ t❤❡ t②♣❡ ❧❛♥❣✉❛❣❡s✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-43
SLIDE 43

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

■❞❡❛✿ L✶ := L❘❊●(P❆❚)✱ L✷ := LL✶(P❆❚)✱ L✸ := LL✷(P❆❚)✱ ✳ ✳ ✳ Pr♦♣♦s✐t✐♦♥ ❋♦r ❛♥② ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s ✱ P❆❚

P❆❚ P❆❚ ✳

❍❡♥❝❡✱ t❤❡ ❛s♣❡❝t ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❝❛♥♥♦t ❜❡ ❧✐♠✐t❡❞ t♦ t❤❡ t②♣❡ ❧❛♥❣✉❛❣❡s✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-44
SLIDE 44

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

■❞❡❛✿ L✶ := L❘❊●(P❆❚)✱ L✷ := LL✶(P❆❚)✱ L✸ := LL✷(P❆❚)✱ ✳ ✳ ✳ Pr♦♣♦s✐t✐♦♥ ❋♦r ❛♥② ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s L✱ LL(P❆❚) = LLL(P❆❚)(P❆❚)✳ ❍❡♥❝❡✱ t❤❡ ❛s♣❡❝t ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❝❛♥♥♦t ❜❡ ❧✐♠✐t❡❞ t♦ t❤❡ t②♣❡ ❧❛♥❣✉❛❣❡s✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-45
SLIDE 45

✭❚②♣❡❞✮ P❛tt❡r♥ ❧❛♥❣✉❛❣❡s

■❞❡❛✿ L✶ := L❘❊●(P❆❚)✱ L✷ := LL✶(P❆❚)✱ L✸ := LL✷(P❆❚)✱ ✳ ✳ ✳ Pr♦♣♦s✐t✐♦♥ ❋♦r ❛♥② ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s L✱ LL(P❆❚) = LLL(P❆❚)(P❆❚)✳ ❍❡♥❝❡✱ t❤❡ ❛s♣❡❝t ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❝❛♥♥♦t ❜❡ ❧✐♠✐t❡❞ t♦ t❤❡ t②♣❡ ❧❛♥❣✉❛❣❡s✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-46
SLIDE 46

P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

P❆❚r♦ := {α | α ✐s ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♦✈❡r (Σ ∪ ❳)}✳ ❊✈❡r② α ∈ P❆❚r♦ ✐s ❛ ♣❛tt❡r♥ ✇✐t❤ r❡❣✉❧❛r ♦♣❡r❛t♦rs✳

✶ ✷ ✸

✱ ✇❤❡r❡ ✐s ❛ t②♣❡ ❢♦r ❛♥❞

✶ ✷ ✸

✳ ❊①❛♠♣❧❡✿

❜

①✶❝

❜

①✶❝

❜

①✶❝①✶❝

❜

①✶❝①✶❝①✶❝ ❜♥❝ ♠ ♥ ✵ ♠ ✶ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-47
SLIDE 47

P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

P❆❚r♦ := {α | α ✐s ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♦✈❡r (Σ ∪ ❳)}✳ ❊✈❡r② α ∈ P❆❚r♦ ✐s ❛ ♣❛tt❡r♥ ✇✐t❤ r❡❣✉❧❛r ♦♣❡r❛t♦rs✳ LT (α) := LT (β✶) ∪ LT (β✷) ∪ LT (β✸) ∪ . . .✱ ✇❤❡r❡ ✐s ❛ t②♣❡ ❢♦r ❛♥❞

✶ ✷ ✸

✳ ❊①❛♠♣❧❡✿

❜

①✶❝

❜

①✶❝

❜

①✶❝①✶❝

❜

①✶❝①✶❝①✶❝ ❜♥❝ ♠ ♥ ✵ ♠ ✶ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-48
SLIDE 48

P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

P❆❚r♦ := {α | α ✐s ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♦✈❡r (Σ ∪ ❳)}✳ ❊✈❡r② α ∈ P❆❚r♦ ✐s ❛ ♣❛tt❡r♥ ✇✐t❤ r❡❣✉❧❛r ♦♣❡r❛t♦rs✳ LT (α) := LT (β✶) ∪ LT (β✷) ∪ LT (β✸) ∪ . . .✱ ✇❤❡r❡ T ✐s ❛ t②♣❡ ❢♦r α ❛♥❞

✶ ✷ ✸

✳ ❊①❛♠♣❧❡✿

❜

①✶❝

❜

①✶❝

❜

①✶❝①✶❝

❜

①✶❝①✶❝①✶❝ ❜♥❝ ♠ ♥ ✵ ♠ ✶ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-49
SLIDE 49

P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

P❆❚r♦ := {α | α ✐s ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♦✈❡r (Σ ∪ ❳)}✳ ❊✈❡r② α ∈ P❆❚r♦ ✐s ❛ ♣❛tt❡r♥ ✇✐t❤ r❡❣✉❧❛r ♦♣❡r❛t♦rs✳ LT (α) := LT (β✶) ∪ LT (β✷) ∪ LT (β✸) ∪ . . .✱ ✇❤❡r❡ T ✐s ❛ t②♣❡ ❢♦r α ❛♥❞ L(α) = {β✶, β✷, β✸, . . .}✳ ❊①❛♠♣❧❡✿

❜

①✶❝

❜

①✶❝

❜

①✶❝①✶❝

❜

①✶❝①✶❝①✶❝ ❜♥❝ ♠ ♥ ✵ ♠ ✶ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-50
SLIDE 50

P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

P❆❚r♦ := {α | α ✐s ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♦✈❡r (Σ ∪ ❳)}✳ ❊✈❡r② α ∈ P❆❚r♦ ✐s ❛ ♣❛tt❡r♥ ✇✐t❤ r❡❣✉❧❛r ♦♣❡r❛t♦rs✳ LT (α) := LT (β✶) ∪ LT (β✷) ∪ LT (β✸) ∪ . . .✱ ✇❤❡r❡ T ✐s ❛ t②♣❡ ❢♦r α ❛♥❞ L(α) = {β✶, β✷, β✸, . . .}✳ ❊①❛♠♣❧❡✿ L(L(❜∗))((①✶❝)+) =

❜

①✶❝

❜

①✶❝①✶❝

❜

①✶❝①✶❝①✶❝ ❜♥❝ ♠ ♥ ✵ ♠ ✶ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-51
SLIDE 51

P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

P❆❚r♦ := {α | α ✐s ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♦✈❡r (Σ ∪ ❳)}✳ ❊✈❡r② α ∈ P❆❚r♦ ✐s ❛ ♣❛tt❡r♥ ✇✐t❤ r❡❣✉❧❛r ♦♣❡r❛t♦rs✳ LT (α) := LT (β✶) ∪ LT (β✷) ∪ LT (β✸) ∪ . . .✱ ✇❤❡r❡ T ✐s ❛ t②♣❡ ❢♦r α ❛♥❞ L(α) = {β✶, β✷, β✸, . . .}✳ ❊①❛♠♣❧❡✿ L(L(❜∗))((①✶❝)+) = L(L(❜∗))(①✶❝) ∪ L(L(❜∗))(①✶❝①✶❝) ∪ L(L(❜∗))(①✶❝①✶❝①✶❝) ∪ . . . = ❜♥❝ ♠ ♥ ✵ ♠ ✶ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-52
SLIDE 52

P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

P❆❚r♦ := {α | α ✐s ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♦✈❡r (Σ ∪ ❳)}✳ ❊✈❡r② α ∈ P❆❚r♦ ✐s ❛ ♣❛tt❡r♥ ✇✐t❤ r❡❣✉❧❛r ♦♣❡r❛t♦rs✳ LT (α) := LT (β✶) ∪ LT (β✷) ∪ LT (β✸) ∪ . . .✱ ✇❤❡r❡ T ✐s ❛ t②♣❡ ❢♦r α ❛♥❞ L(α) = {β✶, β✷, β✸, . . .}✳ ❊①❛♠♣❧❡✿ L(L(❜∗))((①✶❝)+) = L(L(❜∗))(①✶❝) ∪ L(L(❜∗))(①✶❝①✶❝) ∪ L(L(❜∗))(①✶❝①✶❝①✶❝) ∪ . . . = {(❜♥❝)♠ | ♥ ≥ ✵, ♠ ≥ ✶}✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-53
SLIDE 53

❊①♣r❡ss✐✈❡ P♦✇❡r

❚❤❡♦r❡♠ L{Σ∗}(P❆❚) ⊂ L❘❊●(P❆❚) ⊂ L❘❊●(P❆❚r♦)✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-54
SLIDE 54

■t❡r❛t✐✈❡❧② ❚②♣✐♥❣ P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

Lr♦,✵ := ❘❊●✱

r♦ ✶

r♦ ✵ P❆❚r♦

❘❊● P❆❚r♦ ✱ r♦ ✷

r♦ ✶ P❆❚r♦ ✱

r♦ ✸

r♦ ✷ P❆❚r♦ ✱

✳ ✳ ✳

r♦ ✐ ✵ r♦ ✐✳

❚❤❡♦r❡♠

r♦ ✵ r♦ ✶ r♦ ✷ r♦ ✸ r♦ ✹

✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-55
SLIDE 55

■t❡r❛t✐✈❡❧② ❚②♣✐♥❣ P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

Lr♦,✵ := ❘❊●✱ Lr♦,✶ := LLr♦,✵(P❆❚r♦) = L❘❊●(P❆❚r♦)✱

r♦ ✷

r♦ ✶ P❆❚r♦ ✱

r♦ ✸

r♦ ✷ P❆❚r♦ ✱

✳ ✳ ✳

r♦ ✐ ✵ r♦ ✐✳

❚❤❡♦r❡♠

r♦ ✵ r♦ ✶ r♦ ✷ r♦ ✸ r♦ ✹

✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-56
SLIDE 56

■t❡r❛t✐✈❡❧② ❚②♣✐♥❣ P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

Lr♦,✵ := ❘❊●✱ Lr♦,✶ := LLr♦,✵(P❆❚r♦) = L❘❊●(P❆❚r♦)✱ Lr♦,✷ := LLr♦,✶(P❆❚r♦)✱

r♦ ✸

r♦ ✷ P❆❚r♦ ✱

✳ ✳ ✳

r♦ ✐ ✵ r♦ ✐✳

❚❤❡♦r❡♠

r♦ ✵ r♦ ✶ r♦ ✷ r♦ ✸ r♦ ✹

✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-57
SLIDE 57

■t❡r❛t✐✈❡❧② ❚②♣✐♥❣ P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

Lr♦,✵ := ❘❊●✱ Lr♦,✶ := LLr♦,✵(P❆❚r♦) = L❘❊●(P❆❚r♦)✱ Lr♦,✷ := LLr♦,✶(P❆❚r♦)✱ Lr♦,✸ := LLr♦,✷(P❆❚r♦)✱ ✳ ✳ ✳

r♦ ✐ ✵ r♦ ✐✳

❚❤❡♦r❡♠

r♦ ✵ r♦ ✶ r♦ ✷ r♦ ✸ r♦ ✹

✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-58
SLIDE 58

■t❡r❛t✐✈❡❧② ❚②♣✐♥❣ P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

Lr♦,✵ := ❘❊●✱ Lr♦,✶ := LLr♦,✵(P❆❚r♦) = L❘❊●(P❆❚r♦)✱ Lr♦,✷ := LLr♦,✶(P❆❚r♦)✱ Lr♦,✸ := LLr♦,✷(P❆❚r♦)✱ ✳ ✳ ✳ Lr♦,∞ := ∞

✐=✵ Lr♦,✐✳

❚❤❡♦r❡♠

r♦ ✵ r♦ ✶ r♦ ✷ r♦ ✸ r♦ ✹

✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-59
SLIDE 59

■t❡r❛t✐✈❡❧② ❚②♣✐♥❣ P❛tt❡r♥s ✇✐t❤ ❘❡❣✉❧❛r ❖♣❡r❛t♦rs

Lr♦,✵ := ❘❊●✱ Lr♦,✶ := LLr♦,✵(P❆❚r♦) = L❘❊●(P❆❚r♦)✱ Lr♦,✷ := LLr♦,✶(P❆❚r♦)✱ Lr♦,✸ := LLr♦,✷(P❆❚r♦)✱ ✳ ✳ ✳ Lr♦,∞ := ∞

✐=✵ Lr♦,✐✳

❚❤❡♦r❡♠ Lr♦,✵ ⊂ Lr♦,✶ ⊂ Lr♦,✷ ⊆ Lr♦,✸ ⊆ Lr♦,✹ ⊆ . . . ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-60
SLIDE 60

P❛tt❡r♥ ❊①♣r❡ss✐♦♥s

■♥tr♦❞✉❝❡❞ ❜② ❈â♠♣❡❛♥✉ ❛♥❞ ❨✉✱ ✷✵✵✹✳ ❆ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ✐s ❛ t✉♣❧❡ (①✶ → r✶, ①✷ → r✷, . . . , ①♥ → r♥) , ✇❤❡r❡ ✈❛r r✶ ✱ ✈❛r r✷ ①✶ ✱ ✈❛r r✸ ①✶ ①✷ ✱ ✈❛r r✹ ①✶ ①✷ ①✸ ✱ ✳ ✳ ✳ ❚❤❡ s❡t ♦❢ ❛❧❧ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥s ✐s ❞❡♥♦t❡❞ ❜② P❊✳ ❊①❛♠♣❧❡✿ q ①✶ ❛ ①✷ ①✶ ❝ ❞ ①✶ ①✸ ①✶❝①✷ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-61
SLIDE 61

P❛tt❡r♥ ❊①♣r❡ss✐♦♥s

■♥tr♦❞✉❝❡❞ ❜② ❈â♠♣❡❛♥✉ ❛♥❞ ❨✉✱ ✷✵✵✹✳ ❆ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ✐s ❛ t✉♣❧❡ (①✶ → r✶, ①✷ → r✷, . . . , ①♥ → r♥) , ✇❤❡r❡ ✈❛r(r✶) = ∅✱ ✈❛r(r✷) ⊆ {①✶}✱ ✈❛r(r✸) ⊆ {①✶, ①✷}✱ ✈❛r(r✹) ⊆ {①✶, ①✷, ①✸}✱ ✳ ✳ ✳ ❚❤❡ s❡t ♦❢ ❛❧❧ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥s ✐s ❞❡♥♦t❡❞ ❜② P❊✳ ❊①❛♠♣❧❡✿ q ①✶ ❛ ①✷ ①✶ ❝ ❞ ①✶ ①✸ ①✶❝①✷ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-62
SLIDE 62

P❛tt❡r♥ ❊①♣r❡ss✐♦♥s

■♥tr♦❞✉❝❡❞ ❜② ❈â♠♣❡❛♥✉ ❛♥❞ ❨✉✱ ✷✵✵✹✳ ❆ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ✐s ❛ t✉♣❧❡ (①✶ → r✶, ①✷ → r✷, . . . , ①♥ → r♥) , ✇❤❡r❡ ✈❛r(r✶) = ∅✱ ✈❛r(r✷) ⊆ {①✶}✱ ✈❛r(r✸) ⊆ {①✶, ①✷}✱ ✈❛r(r✹) ⊆ {①✶, ①✷, ①✸}✱ ✳ ✳ ✳ ❚❤❡ s❡t ♦❢ ❛❧❧ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥s ✐s ❞❡♥♦t❡❞ ❜② P❊✳ ❊①❛♠♣❧❡✿ q ①✶ ❛ ①✷ ①✶ ❝ ❞ ①✶ ①✸ ①✶❝①✷ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-63
SLIDE 63

P❛tt❡r♥ ❊①♣r❡ss✐♦♥s

■♥tr♦❞✉❝❡❞ ❜② ❈â♠♣❡❛♥✉ ❛♥❞ ❨✉✱ ✷✵✵✹✳ ❆ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ✐s ❛ t✉♣❧❡ (①✶ → r✶, ①✷ → r✷, . . . , ①♥ → r♥) , ✇❤❡r❡ ✈❛r(r✶) = ∅✱ ✈❛r(r✷) ⊆ {①✶}✱ ✈❛r(r✸) ⊆ {①✶, ①✷}✱ ✈❛r(r✹) ⊆ {①✶, ①✷, ①✸}✱ ✳ ✳ ✳ ❚❤❡ s❡t ♦❢ ❛❧❧ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥s ✐s ❞❡♥♦t❡❞ ❜② P❊✳ ❊①❛♠♣❧❡✿ q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-64
SLIDE 64

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛ ①✶ ❝ ❞ ①✶ ①✶ ❝ ①✷

✐t q

❛❦❝❛♠✉❛♠ ❦ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-65
SLIDE 65

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ ①✶ (❝ | ❞) ①✶ ①✶ ❝ ①✷

✐t q

❛❦❝❛♠✉❛♠ ❦ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-66
SLIDE 66

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ ①✶ (❝ | ❞) ①✶ ①✶ ❝ ①✷

✐t q

❛❦❝❛♠✉❛♠ ❦ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-67
SLIDE 67

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ①✶ ❝ ①✷ ❛❛❛ ❛❛❛

✐t q

❛❦❝❛♠✉❛♠ ❦ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 68

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ①✶ ❝ ①✷ ❛❛❛ ❛❛❛

✐t q

❛❦❝❛♠✉❛♠ ❦ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

slide-69
SLIDE 69

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ①✶ ❝ ❛❛❛ ❛❛❛ ❛❛❛❝❛❛❛

✐t q

❛❦❝❛♠✉❛♠ ❦ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 70

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ①✶ ❝ ❛❛❛ ❛❛❛ ❛❛❛❝❛❛❛

✐t q

❛❦❝❛♠✉❛♠ ❦ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 71

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ❝ ❛❛❛ ❛❛❛ ❛ ❛❛❛❝❛❛❛

✐t q

❛❦❝❛♠✉❛♠ ❦ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 72

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ❝ ❛❛❛ ❛❛❛ ❛ ❛❛❛❝❛❛❛

✐t q

❛❦❝❛♠✉❛♠ ❦ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 73

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ❝ ❛❛❛ ❛❛❛ ❛ ❛❛❛❝❛❛❛ L✐t(q) = {❛❦❝❛♠✉❛♠ | ❦, ♠ ∈ N✵, ✉ ∈ {❝, ❞}} ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✐t❡r❛t❡❞ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 74

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛ ①✶ ❝ ❞ ①✶ ①✶ ❝ ①✷

✉♥✐ q

❛♠❝❛♠✉❛♠ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✉♥✐❢♦r♠ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 75

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ ①✶ (❝ | ❞) ①✶ ①✶ ❝ ①✷

✉♥✐ q

❛♠❝❛♠✉❛♠ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✉♥✐❢♦r♠ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 76

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ ①✶ (❝ | ❞) ①✶ ①✶ ❝ ①✷

✉♥✐ q

❛♠❝❛♠✉❛♠ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✉♥✐❢♦r♠ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 77

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ❝ ①✷ ❛❛❛ ❛❛❛ ❛❛❛

✉♥✐ q

❛♠❝❛♠✉❛♠ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✉♥✐❢♦r♠ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 78

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ❝ ①✷ ❛❛❛ ❛❛❛ ❛❛❛

✉♥✐ q

❛♠❝❛♠✉❛♠ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✉♥✐❢♦r♠ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 79

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ❝ ❛❛❛ ❛❛❛ ❛❛❛ ❛❛❛❞❛❛❛

✉♥✐ q

❛♠❝❛♠✉❛♠ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✉♥✐❢♦r♠ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 80

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ❝ ❛❛❛ ❛❛❛ ❛❛❛ ❛❛❛❞❛❛❛

✉♥✐ q

❛♠❝❛♠✉❛♠ ♠

✵ ✉

❝ ❞ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✉♥✐❢♦r♠ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 81

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

q := (①✶ → ❛∗, ①✷ → ①✶(❝ | ❞)①✶, ①✸ → ①✶❝①✷)✱ ❛∗ (❝ | ❞) ❝ ❛❛❛ ❛❛❛ ❛❛❛ ❛❛❛❞❛❛❛ L✉♥✐(q) = {❛♠❝❛♠✉❛♠ | ♠ ∈ N✵, ✉ ∈ {❝, ❞}} ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❣❡♥❡r❛t❡❞ ❜② q ✇✐t❤ r❡s♣❡❝t t♦ ✉♥✐❢♦r♠ s✉❜st✐t✉t✐♦♥✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 82

P❛tt❡r♥ ❊①♣r❡ss✐♦♥ ▲❛♥❣✉❛❣❡s

Pr♦♣♦s✐t✐♦♥ ❬❈❛♠♣❡❛♥✉ ❛♥❞ ❨✉❪ ❋♦r ❡✈❡r② ♣ ∈ P❊✱ L✐t(♣) ✐s ❛ ❘❊●❊❳ ❧❛♥❣✉❛❣❡✳ ❚❤❡♦r❡♠ Lr♦,∞ = L✐t(P❊)✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 83

■t❡r❛t❡❞ ✈s✳ ❯♥✐❢♦r♠ ❙✉❜st✐t✉t✐♦♥

Pr♦♣♦s✐t✐♦♥ ▲❡t ♣ := (①✶ → r✶, . . . , ①♠ → r♠) ∈ P❊✳ L✉♥✐(♣) ⊆ L✐t(♣)✱ ✐❢✱ ❢♦r ❡✈❡r② ✐, ❥✱ ✶ ≤ ✐ < ❥ ≤ ♠✱ ✈❛r(r✐) ∩ ✈❛r(r❥) = ∅✱ t❤❡♥ L✐t(♣) ⊆ L✉♥✐(♣)✳ ❚❤❡♦r❡♠

✐t P❊ ✉♥✐ P❊ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 84

■t❡r❛t❡❞ ✈s✳ ❯♥✐❢♦r♠ ❙✉❜st✐t✉t✐♦♥

Pr♦♣♦s✐t✐♦♥ ▲❡t ♣ := (①✶ → r✶, . . . , ①♠ → r♠) ∈ P❊✳ L✉♥✐(♣) ⊆ L✐t(♣)✱ ✐❢✱ ❢♦r ❡✈❡r② ✐, ❥✱ ✶ ≤ ✐ < ❥ ≤ ♠✱ ✈❛r(r✐) ∩ ✈❛r(r❥) = ∅✱ t❤❡♥ L✐t(♣) ⊆ L✉♥✐(♣)✳ ❚❤❡♦r❡♠ L✐t(P❊) ⊂ L✉♥✐(P❊)✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 85

◆♦t❛t✐♦♥

❆ ❘❊●❊❳ r ✐s st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ✐✛ ❡✈❡r② r❡❢❡r❡♥❝❡❞ s✉❜❡①♣r❡ss✐♦♥ ❞♦❡s ♥♦t ♦❝❝✉r ✉♥❞❡r ❛ st❛r✳

✶ ❛

❜

✶❜ ✶

❜ ✶

✶ ❛

❜

✶ ✶ ✷❝ ✷ ❞ ✶ ✷

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 86

◆♦t❛t✐♦♥

❆ ❘❊●❊❳ r ✐s st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ✐✛ ❡✈❡r② r❡❢❡r❡♥❝❡❞ s✉❜❡①♣r❡ss✐♦♥ ❞♦❡s ♥♦t ♦❝❝✉r ✉♥❞❡r ❛ st❛r✳ ((✶(❛ | ❜)∗)✶❜\✶)∗❜\✶ (✶(❛ | ❜)∗)✶\✶(✷❝∗)✷(❞\✶\✷)∗

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 87

P❊ ✇✳ r✳ t✳ ✉♥✐❢♦r♠ s✉❜st✳ ✈s✳ st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ❘❊●❊❳

▲❡♠♠❛ ❋♦r ❡✈❡r② ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ♣✱ t❤❡r❡ ❡①✐sts ❛ st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ❘❊●❊❳ r ✇✐t❤ L✉♥✐(♣) = L(r)✳ ▲❡♠♠❛ ❋♦r ❡✈❡r② st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ❘❊●❊❳ r✱ t❤❡r❡ ❡①✐sts ❛ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ♣ ✇✐t❤ r

✉♥✐ ♣ ✳

❚❤❡♦r❡♠ ❘❊●❊❳s✜

✉♥✐ P❊ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 88

P❊ ✇✳ r✳ t✳ ✉♥✐❢♦r♠ s✉❜st✳ ✈s✳ st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ❘❊●❊❳

▲❡♠♠❛ ❋♦r ❡✈❡r② ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ♣✱ t❤❡r❡ ❡①✐sts ❛ st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ❘❊●❊❳ r ✇✐t❤ L✉♥✐(♣) = L(r)✳ ▲❡♠♠❛ ❋♦r ❡✈❡r② st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ❘❊●❊❳ r✱ t❤❡r❡ ❡①✐sts ❛ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ♣ ✇✐t❤ L(r) = L✉♥✐(♣)✳ ❚❤❡♦r❡♠ ❘❊●❊❳s✜

✉♥✐ P❊ ✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 89

P❊ ✇✳ r✳ t✳ ✉♥✐❢♦r♠ s✉❜st✳ ✈s✳ st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ❘❊●❊❳

▲❡♠♠❛ ❋♦r ❡✈❡r② ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ♣✱ t❤❡r❡ ❡①✐sts ❛ st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ❘❊●❊❳ r ✇✐t❤ L✉♥✐(♣) = L(r)✳ ▲❡♠♠❛ ❋♦r ❡✈❡r② st❛r✲❢r❡❡ ✐♥✐t✐❛❧✐s❡❞ ❘❊●❊❳ r✱ t❤❡r❡ ❡①✐sts ❛ ♣❛tt❡r♥ ❡①♣r❡ss✐♦♥ ♣ ✇✐t❤ L(r) = L✉♥✐(♣)✳ ❚❤❡♦r❡♠ L(❘❊●❊❳s✜) = L✉♥✐(P❊)✳

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 90

L{Σ∗}(P❆❚) ❘❊● L✉♥✐(P❊) L(❘❊●❊❳s✜) L(❘❊●❊❳) L✐t(P❊) H∗(❘❊●, ❘❊●) Lr♦,∞ Lr♦,✶ Lr♦,✷ Lr♦,✸ ✳ ✳ ✳ L❘❊●(P❆❚) s✉❜s❡t ♣r♦♣❡r s✉❜s❡t ❡q✉❛❧✐t②

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 91

L❘❊●(P❆❚) L{Σ∗}(P❆❚) ❘❊● L✉♥✐(P❊) L(❘❊●❊❳s✜) L(❘❊●❊❳) L✐t(P❊) H∗(❘❊●, ❘❊●) Lr♦,∞ Lr♦,✶ Lr♦,✷ Lr♦,✸ ✳ ✳ ✳ s✉❜s❡t ♣r♦♣❡r s✉❜s❡t ❡q✉❛❧✐t②

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 92

Lr♦,∞ Lr♦,✶ Lr♦,✷ Lr♦,✸ ✳ ✳ ✳ L❘❊●(P❆❚) L{Σ∗}(P❆❚) ❘❊● L✉♥✐(P❊) L(❘❊●❊❳s✜) L(❘❊●❊❳) L✐t(P❊) H∗(❘❊●, ❘❊●) s✉❜s❡t ♣r♦♣❡r s✉❜s❡t ❡q✉❛❧✐t②

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 93

L✐t(P❊) Lr♦,∞ Lr♦,✶ Lr♦,✷ Lr♦,✸ ✳ ✳ ✳ L❘❊●(P❆❚) L{Σ∗}(P❆❚) ❘❊● L✉♥✐(P❊) L(❘❊●❊❳s✜) L(❘❊●❊❳) H∗(❘❊●, ❘❊●) s✉❜s❡t ♣r♦♣❡r s✉❜s❡t ❡q✉❛❧✐t②

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 94

L✐t(P❊) H∗(❘❊●, ❘❊●) Lr♦,∞ Lr♦,✶ Lr♦,✷ Lr♦,✸ ✳ ✳ ✳ L❘❊●(P❆❚) L{Σ∗}(P❆❚) ❘❊● L✉♥✐(P❊) L(❘❊●❊❳s✜) L(❘❊●❊❳) s✉❜s❡t ♣r♦♣❡r s✉❜s❡t ❡q✉❛❧✐t② ❆❧❜❡rt✱ ❲❡❣♥❡r✱ ✶✾✽✶ ❇♦r❞✐❤♥ ❡t ❛❧✳✱ ✷✵✶✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 95

L✉♥✐(P❊) L✐t(P❊) H∗(❘❊●, ❘❊●) Lr♦,∞ Lr♦,✶ Lr♦,✷ Lr♦,✸ ✳ ✳ ✳ L❘❊●(P❆❚) L{Σ∗}(P❆❚) ❘❊● L(❘❊●❊❳s✜) L(❘❊●❊❳) s✉❜s❡t ♣r♦♣❡r s✉❜s❡t ❡q✉❛❧✐t② ❆❧❜❡rt✱ ❲❡❣♥❡r✱ ✶✾✽✶ ❇♦r❞✐❤♥ ❡t ❛❧✳✱ ✷✵✶✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 96

L✉♥✐(P❊) L(❘❊●❊❳s✜) L✐t(P❊) H∗(❘❊●, ❘❊●) Lr♦,∞ Lr♦,✶ Lr♦,✷ Lr♦,✸ ✳ ✳ ✳ L❘❊●(P❆❚) L{Σ∗}(P❆❚) ❘❊● L(❘❊●❊❳) s✉❜s❡t ♣r♦♣❡r s✉❜s❡t ❡q✉❛❧✐t② ❆❧❜❡rt✱ ❲❡❣♥❡r✱ ✶✾✽✶ ❇♦r❞✐❤♥ ❡t ❛❧✳✱ ✷✵✶✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s

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SLIDE 97

L✉♥✐(P❊) L(❘❊●❊❳s✜) L(❘❊●❊❳) L✐t(P❊) H∗(❘❊●, ❘❊●) Lr♦,∞ Lr♦,✶ Lr♦,✷ Lr♦,✸ ✳ ✳ ✳ L❘❊●(P❆❚) L{Σ∗}(P❆❚) ❘❊● s✉❜s❡t ♣r♦♣❡r s✉❜s❡t ❡q✉❛❧✐t② ❆❧❜❡rt✱ ❲❡❣♥❡r✱ ✶✾✽✶ ❇♦r❞✐❤♥ ❡t ❛❧✳✱ ✷✵✶✵

▼❛r❦✉s ▲✳ ❙❝❤♠✐❞✱ ▲♦✉❣❤❜♦r♦✉❣❤ ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥s✐❞❡ t❤❡ ❈❧❛ss ♦❢ ❘❊●❊❳ ▲❛♥❣✉❛❣❡s