Formalising Regular Language Theory with Regular Expressions, Only - - PowerPoint PPT Presentation

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Formalising Regular Language Theory with Regular Expressions, Only - - PowerPoint PPT Presentation

Aug 01, 2023 204 likes •769 views

Formalising Regular Language Theory with Regular Expressions, Only rst r s t r t

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SLIDE 1

Formalising Regular Language Theory with Regular Expressions, Only

❈❤r✐st✐❛♥ ❯r❜❛♥

❑✐♥❣✬s ❈♦❧❧❡❣❡ ▲♦♥❞♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❈❤✉♥❤❛♥ ❲✉ ❛♥❞ ❳✐♥❣②✉❛♥ ❩❤❛♥❣ ❢r♦♠ t❤❡ P▲❆ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ✐♥ ◆❛♥❥✐♥❣

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✴✶✽

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SLIDE 2

Formalising Regular Language Theory with Regular Expressions, Only

❈❤r✐st✐❛♥ ❯r❜❛♥

❑✐♥❣✬s ❈♦❧❧❡❣❡ ▲♦♥❞♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❈❤✉♥❤❛♥ ❲✉ ❛♥❞ ❳✐♥❣②✉❛♥ ❩❤❛♥❣ ❢r♦♠ t❤❡ P▲❆ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ✐♥ ◆❛♥❥✐♥❣

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✴✶✽

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SLIDE 3

❘♦② ✐♥t❡rt✇✐♥❡❞ ✇✐t❤ ♠② s❝✐❡♥t✐✂❝ ❧✐❢❡ ♦♥ ♠❛♥② ♦❝❝❛s✐♦♥s✱ ♠♦st ♥♦t❛❜❧②✿ ❤❡ ❛❞♠✐tt❡❞ ♠❡ ❢♦r ▼✳P❤✐❧✳ ✐♥ ❙t ❆♥❞r❡✇s ❛♥❞ ♠❛❞❡ ♠❡ ❧✐❦❡ t❤❡♦r② s❡♥t ♠❡ t♦ ❈❛♠❜r✐❞❣❡ ❢♦r P❤✳❉✳ ♠❛❞❡ ♠❡ ❛♣♣r❡❝✐❛t❡ ♣r❡❝✐s✐♦♥ ✐♥ ♣r♦♦❢s

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✷✴✶✽

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SLIDE 4

❇♦❜ ❍❛r♣❡r ✭❈▼❯✮ ❋r❛♥❦ P❢❡♥♥✐♥❣ ✭❈▼❯✮

♣✉❜❧✐s❤❡❞ ❛ ♣r♦♦❢ ✐♥ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲♦❣✐❝✱ ✷✵✵✺✱

∼✸✶♣♣

❆♥❞r❡✇ ❆♣♣❡❧ ✭Pr✐♥❝❡t♦♥✮

r❡❧✐❡❞ ♦♥ t❤❡✐r ♣r♦♦❢ ✐♥ ❛ s❡❝✉r✐t② ❝r✐t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥

✭■ ❛❧s♦ ❢♦✉♥❞ ❛♥ ❡rr♦r ✐♥ ♠② P❤✳❉✳✲t❤❡s✐s ❛❜♦✉t ❝✉t✲❡❧✐♠✐♥❛t✐♦♥ ❡①❛♠✐♥❡❞ ❜② ❍❡♥❦ ❇❛r❡♥❞r❡❣t ❛♥❞ ❆♥❞② P✐tts✳✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✸✴✶✽

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SLIDE 5

❇♦❜ ❍❛r♣❡r ✭❈▼❯✮ ❋r❛♥❦ P❢❡♥♥✐♥❣ ✭❈▼❯✮

♣✉❜❧✐s❤❡❞ ❛ ♣r♦♦❢ ✐♥ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲♦❣✐❝✱ ✷✵✵✺✱

∼✸✶♣♣

❆♥❞r❡✇ ❆♣♣❡❧ ✭Pr✐♥❝❡t♦♥✮

r❡❧✐❡❞ ♦♥ t❤❡✐r ♣r♦♦❢ ✐♥ ❛ s❡❝✉r✐t② ❝r✐t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥

✭■ ❛❧s♦ ❢♦✉♥❞ ❛♥ ❡rr♦r ✐♥ ♠② P❤✳❉✳✲t❤❡s✐s ❛❜♦✉t ❝✉t✲❡❧✐♠✐♥❛t✐♦♥ ❡①❛♠✐♥❡❞ ❜② ❍❡♥❦ ❇❛r❡♥❞r❡❣t ❛♥❞ ❆♥❞② P✐tts✳✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✸✴✶✽

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SLIDE 6

❇♦❜ ❍❛r♣❡r ✭❈▼❯✮ ❋r❛♥❦ P❢❡♥♥✐♥❣ ✭❈▼❯✮

♣✉❜❧✐s❤❡❞ ❛ ♣r♦♦❢ ✐♥ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲♦❣✐❝✱ ✷✵✵✺✱

∼✸✶♣♣

❆♥❞r❡✇ ❆♣♣❡❧ ✭Pr✐♥❝❡t♦♥✮

r❡❧✐❡❞ ♦♥ t❤❡✐r ♣r♦♦❢ ✐♥ ❛ s❡❝✉r✐t② ❝r✐t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥

✭■ ❛❧s♦ ❢♦✉♥❞ ❛♥ ❡rr♦r ✐♥ ♠② P❤✳❉✳✲t❤❡s✐s ❛❜♦✉t ❝✉t✲❡❧✐♠✐♥❛t✐♦♥ ❡①❛♠✐♥❡❞ ❜② ❍❡♥❦ ❇❛r❡♥❞r❡❣t ❛♥❞ ❆♥❞② P✐tts✳✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✸✴✶✽

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SLIDE 7

Formal language theory.. .

✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs

❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳

❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

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SLIDE 8

Formal language theory.. .

✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs

❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳

❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s

A1 A2

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

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SLIDE 9

Formal language theory.. .

✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs

❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳

❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s

A1 A2

⇒

A1 A2

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

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SLIDE 10

Formal language theory.. .

✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs

❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳

❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s

A1 A2

⇒

A1 A2

❞✐s❥♦✐♥t ✉♥✐♦♥✿

A1 ⊎ A2

❞❡❢

= {(1, x) | x ∈ A1} ∪ {(2, y) | y ∈ A2}

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

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SLIDE 11

Formal language theory.. .

✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs

❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳

❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s

A1 A2

⇒

A1 A2

❞✐s❥♦✐♥t ✉♥✐♦♥✿

A1 ⊎ A2

❞❡❢

= {(1, x) | x ∈ A1} ∪ {(2, y) | y ∈ A2}

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

Pr♦❜❧❡♠s ✇✐t❤ ❞❡✂♥✐t✐♦♥ ❢♦r r❡❣✉❧❛r✐t②✿ ✐s❴r❡❣✉❧❛r(A)

❞❡❢

= ∃M. ✐s❴❞❢❛(M) ∧ L(M) = A

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SLIDE 12

Formal language theory.. .

✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs

❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳

❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s

A1 A2

⇒

A1 A2

❆ s♦❧✉t✐♦♥✿ ✉s❡ ♥❛ts ⇒ st❛t❡ ♥♦❞❡s

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

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SLIDE 13

Formal language theory.. .

✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs

❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳

❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s

A1 A2

⇒

A1 A2

❆ s♦❧✉t✐♦♥✿ ✉s❡ ♥❛ts ⇒ st❛t❡ ♥♦❞❡s ❨♦✉ ❤❛✈❡ t♦ r❡♥❛♠❡ st❛t❡s✦

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

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SLIDE 14

Formal language theory.. .

✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs

❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳

❑♦③❡♥✬s ➇♣❛♣❡r➈ ♣r♦♦❢ ♦❢ ▼②❤✐❧❧✲◆❡r♦❞❡✿ r❡q✉✐r❡s ❛❜s❡♥❝❡ ♦❢ ✐♥❛❝❝❡ss✐❜❧❡ st❛t❡s ✐s❴r❡❣✉❧❛r(A)

❞❡❢

= ∃M. ✐s❴❞❢❛(M) ∧ L(M) = A

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✺✴✶✽

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SLIDE 15

✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛

■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A✳

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SLIDE 16

✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛

■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A✳

slide-17
SLIDE 17

✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛

■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A✳

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SLIDE 18

✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛

■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A✳

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SLIDE 19

✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛

■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A✳

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SLIDE 20

✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛

■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♠❛t❝❤✐♥❣ ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A✳

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SLIDE 21

✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛

■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♠❛t❝❤✐♥❣ ✭⇒❇r♦③♦✇s❦✐✬✻✹✱ ❖✇❡♥s ❡t ❛❧ ✬✵✾✮ ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A✳

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SLIDE 22

✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛

■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ ♠❛t❝❤✐♥❣ ✭⇒❇r♦③♦✇s❦✐✬✻✹✱ ❖✇❡♥s ❡t ❛❧ ✬✵✾✮ ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A✳

slide-23
SLIDE 23

The Myhill-Nerode Theorem

♣r♦✈✐❞❡s ♥❡❝❡ss❛r② ❛♥❞ s✉❢ ✂❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r ❛ ❧❛♥❣✉❛❣❡ ❜❡✐♥❣ r❡❣✉❧❛r ✭♣✉♠♣✐♥❣ ❧❡♠♠❛ ♦♥❧② ♥❡❝❡ss❛r②✮ ❦❡② ✐s t❤❡ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✿

x ≈A y

❞❡❢

= ∀z. x@z ∈ A ⇔ y@z ∈ A

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✼✴✶✽

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SLIDE 24

The Myhill-Nerode Theorem

✂♥✐t❡ (UNIV/

/ ≈A) ⇔ A ✐s r❡❣✉❧❛r

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✽✴✶✽

UNIV

s❡t ♦❢ ❛❧❧ str✐♥❣s

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SLIDE 25

The Myhill-Nerode Theorem

✂♥✐t❡ (UNIV/

/ ≈A) ⇔ A ✐s r❡❣✉❧❛r

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✽✴✶✽

UNIV

s❡t ♦❢ ❛❧❧ str✐♥❣s

[ [①] ]≈A

❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss

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SLIDE 26

The Myhill-Nerode Theorem

✂♥✐t❡ (UNIV/

/ ≈A) ⇔ A ✐s r❡❣✉❧❛r

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✽✴✶✽

UNIV

s❡t ♦❢ ❛❧❧ str✐♥❣s

[ [①] ]≈A

❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss ❚✇♦ ❞✐r❡❝t✐♦♥s✿ ✶✳✮ ✂♥✐t❡ ⇒ r❡❣✉❧❛r ✂♥✐t❡ (UNIV/

/ ≈A) ⇒ ∃r. A = L(r)

✷✳✮ r❡❣✉❧❛r ⇒ ✂♥✐t❡ ✂♥✐t❡ (UNIV/

/ ≈L(r))

slide-27
SLIDE 27

Initial and Final States

✂♥❛❧s A

❞❡❢

= {[ |x| ]≈A | x ∈ A}

✇❡ ❝❛♥ ♣r♦✈❡✿ A = ✂♥❛❧s A

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✾✴✶✽

❊q✉✐✈❛❧❡♥❝❡ ❈❧❛ss❡s

slide-28
SLIDE 28

Initial and Final States

✂♥❛❧s A

❞❡❢

= {[ |x| ]≈A | x ∈ A}

✇❡ ❝❛♥ ♣r♦✈❡✿ A = ✂♥❛❧s A

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✾✴✶✽

❊q✉✐✈❛❧❡♥❝❡ ❈❧❛ss❡s

[] ∈ X

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SLIDE 29

Initial and Final States

✂♥❛❧s A

❞❡❢

= {[ |x| ]≈A | x ∈ A}

✇❡ ❝❛♥ ♣r♦✈❡✿ A = ✂♥❛❧s A

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✾✴✶✽

❊q✉✐✈❛❧❡♥❝❡ ❈❧❛ss❡s

[] ∈ X

❛ ✂♥❛❧

slide-30
SLIDE 30

Transitions between Eq-Classes

X Y c X

c

− → Y

❞❡❢

= X; c ⊆ Y

st❛rt

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✵✴✶✽

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SLIDE 31

Systems of Equations

■♥s♣✐r❡❞ ❜② ❛ ♠❡t❤♦❞ ♦❢ ❇r③♦③♦✇s❦✐ ✬✻✹✿

X1

st❛rt

X2

❛ ❜ ❛ ❜

X1 = X1; b + X2; b X2 = X1; a + X2; a

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✶✴✶✽

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SLIDE 32

Systems of Equations

■♥s♣✐r❡❞ ❜② ❛ ♠❡t❤♦❞ ♦❢ ❇r③♦③♦✇s❦✐ ✬✻✹✿

X1

st❛rt

X2

❛ ❜ ❛ ❜

X1 = X1; b + X2; b + λ; [] X2 = X1; a + X2; a

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✶✴✶✽

slide-33
SLIDE 33

X1 = X1; b + X2; b + λ; [] X2 = X1; a + X2; a

❜② ❆r❞❡♥ ❜② ❆r❞❡♥ ❜② s✉❜st✐t✉t✐♦♥ ❜② ❆r❞❡♥ ❜② s✉❜st✐t✉t✐♦♥

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✷✴✶✽

❛

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SLIDE 34

X1 = X1; b + X2; b + λ; [] X2 = X1; a + X2; a

❜② ❆r❞❡♥

X1 = X1; b + X2; b + λ; [] X2 = X1; a · a⋆

❜② ❆r❞❡♥ ❜② s✉❜st✐t✉t✐♦♥ ❜② ❆r❞❡♥ ❜② s✉❜st✐t✉t✐♦♥

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✷✴✶✽

❛ ❛

slide-35
SLIDE 35

X1 = X1; b + X2; b + λ; [] X2 = X1; a + X2; a

❜② ❆r❞❡♥

X1 = X1; b + X2; b + λ; [] X2 = X1; a · a⋆

❜② ❆r❞❡♥

X1 = X2; b · b⋆ + λ; b⋆ X2 = X1; a · a⋆

❜② s✉❜st✐t✉t✐♦♥ ❜② ❆r❞❡♥ ❜② s✉❜st✐t✉t✐♦♥

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✷✴✶✽

❛ ❛

slide-36
SLIDE 36

X1 = X1; b + X2; b + λ; [] X2 = X1; a + X2; a

❜② ❆r❞❡♥

X1 = X1; b + X2; b + λ; [] X2 = X1; a · a⋆

❜② ❆r❞❡♥

X1 = X2; b · b⋆ + λ; b⋆ X2 = X1; a · a⋆

❜② s✉❜st✐t✉t✐♦♥

X1 = X1; a · a⋆ · b · b⋆ + λ; b⋆ X2 = X1; a · a⋆

❜② ❆r❞❡♥ ❜② s✉❜st✐t✉t✐♦♥

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✷✴✶✽

❛

slide-37
SLIDE 37

X1 = X1; b + X2; b + λ; [] X2 = X1; a + X2; a

❜② ❆r❞❡♥

X1 = X1; b + X2; b + λ; [] X2 = X1; a · a⋆

❜② ❆r❞❡♥

X1 = X2; b · b⋆ + λ; b⋆ X2 = X1; a · a⋆

❜② s✉❜st✐t✉t✐♦♥

X1 = X1; a · a⋆ · b · b⋆ + λ; b⋆ X2 = X1; a · a⋆

❜② ❆r❞❡♥

X1 = λ; b⋆ · (a · a⋆ · b · b⋆)⋆ X2 = X1; a · a⋆

❜② s✉❜st✐t✉t✐♦♥

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✷✴✶✽

❛ ❛

slide-38
SLIDE 38

X1 = X1; b + X2; b + λ; [] X2 = X1; a + X2; a

❜② ❆r❞❡♥

X1 = X1; b + X2; b + λ; [] X2 = X1; a · a⋆

❜② ❆r❞❡♥

X1 = X2; b · b⋆ + λ; b⋆ X2 = X1; a · a⋆

❜② s✉❜st✐t✉t✐♦♥

X1 = X1; a · a⋆ · b · b⋆ + λ; b⋆ X2 = X1; a · a⋆

❜② ❆r❞❡♥

X1 = λ; b⋆ · (a · a⋆ · b · b⋆)⋆ X2 = X1; a · a⋆

❜② s✉❜st✐t✉t✐♦♥

X1 = λ; b⋆ · (a · a⋆ · b · b⋆)⋆ X2 = λ; b⋆ · (a · a⋆ · b · b⋆)⋆ · a · a⋆

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✷✴✶✽

slide-39
SLIDE 39

X1 = X1; b + X2; b + λ; [] X2 = X1; a + X2; a

❜② ❆r❞❡♥

X1 = X1; b + X2; b + λ; [] X2 = X1; a · a⋆

❜② ❆r❞❡♥

X1 = X2; b · b⋆ + λ; b⋆ X2 = X1; a · a⋆

❜② s✉❜st✐t✉t✐♦♥

X1 = X1; a · a⋆ · b · b⋆ + λ; b⋆ X2 = X1; a · a⋆

❜② ❆r❞❡♥

X1 = λ; b⋆ · (a · a⋆ · b · b⋆)⋆ X2 = X1; a · a⋆

❜② s✉❜st✐t✉t✐♦♥

X1 = λ; b⋆ · (a · a⋆ · b · b⋆)⋆ X2 = λ; b⋆ · (a · a⋆ · b · b⋆)⋆ · a · a⋆

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✷✴✶✽

X1

st❛rt

X2

❛ ❜ ❛ ❜

slide-40
SLIDE 40

The Other Direction

❖♥❡ ❤❛s t♦ ♣r♦✈❡ ✂♥✐t❡(UNIV/

/ ≈L(r))

❜② ✐♥❞✉❝t✐♦♥ ♦♥ r✳ ◆♦t tr✐✈✐❛❧✱ ❜✉t ❛❢t❡r ❛ ❜✐t ♦❢ t❤✐♥❦✐♥❣✱ ♦♥❡ ❝❛♥ ✂♥❞ ❛ r❡✂♥❡❞ r❡❧❛t✐♦♥✿

a1 a2 a3 a4

a1.1 a1.2 a2.1 a2.2 a3.1 a3.2 a4.1 a4.2

UNIV UNIV/ / ≈L(r) UNIV/ /R

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✸✴✶✽

❛

slide-41
SLIDE 41

Derivatives of RExps

✐♥tr♦❞✉❝❡❞ ❜② ❇r♦③♦✇s❦✐ ✬✻✹ ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛❢t❡r ❛ ❝❤❛r❛❝t❡r ❤❛s ❜❡❡♥ ♣❛rs❡❞ ❞❡r ❝ ∅

❞❡❢

= ∅

❞❡r ❝ ❬❪

❞❡❢

= ∅

❞❡r ❝ ❞

❞❡❢

= ✐❢ ❝ = ❞ t❤❡♥ ❬❪ ❡❧s❡ ∅

❞❡r ❝ ✭r1 + r2✮

❞❡❢

= ✭❞❡r ❝ r1✮ + ✭❞❡r ❝ r2✮

❞❡r ❝ ✭r⋆✮

❞❡❢

= ✭❞❡r ❝ r✮ · r⋆

❞❡r ❝ ✭r1 · r2✮

❞❡❢

= ✐❢ ♥✉❧❧❛❜❧❡ r1

t❤❡♥ ✭❞❡r ❝ r1✮ · r2 + ✭❞❡r ❝ r2✮ ❡❧s❡ ✭❞❡r ❝ r1✮ · r2

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✹✴✶✽

slide-42
SLIDE 42

Derivatives of RExps

✐♥tr♦❞✉❝❡❞ ❜② ❇r♦③♦✇s❦✐ ✬✻✹ ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛❢t❡r ❛ ❝❤❛r❛❝t❡r ❤❛s ❜❡❡♥ ♣❛rs❡❞ ♣❞❡r ❝ ∅

❞❡❢

= {}

♣❞❡r ❝ ❬❪

❞❡❢

= {}

♣❞❡r ❝ ❞

❞❡❢

= ✐❢ ❝ = ❞ t❤❡♥ {❬❪} ❡❧s❡ {}

♣❞❡r ❝ ✭r1 + r2✮

❞❡❢

= ✭♣❞❡r ❝ r1✮ ∪ ✭❞❡r ❝ r2✮

♣❞❡r ❝ ✭r⋆✮

❞❡❢

= ✭♣❞❡r ❝ r✮ · r⋆

♣❞❡r ❝ ✭r1 · r2✮

❞❡❢

= ✐❢ ♥✉❧❧❛❜❧❡ r1

t❤❡♥ ✭♣❞❡r ❝ r1✮ · r2 ∪ ✭♣❞❡r ❝ r2✮ ❡❧s❡ ✭♣❞❡r ❝ r1✮ · r2

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✹✴✶✽

♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ❜② ❆♥t✐♠✐r♦✈ ✬✾✺

slide-43
SLIDE 43

Partial Derivatives

♣❞❡rs x r = ♣❞❡rs y r r❡✂♥❡s x ≈L(r) y ✂♥✐t❡ ❚❤❡r❡❢♦r❡ ✂♥✐t❡ ✳ ◗❡❞✳

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✺✴✶✽

slide-44
SLIDE 44

Partial Derivatives

♣❞❡rs x r = ♣❞❡rs y r

  • R

r❡✂♥❡s x ≈L(r) y ✂♥✐t❡(UNIV/

/R)

❚❤❡r❡❢♦r❡ ✂♥✐t❡ ✳ ◗❡❞✳

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✺✴✶✽

❛ ❆♥t✐♠✐r♦✈ ✬✾✺

slide-45
SLIDE 45

Partial Derivatives

♣❞❡rs x r = ♣❞❡rs y r

  • R

r❡✂♥❡s x ≈L(r) y ✂♥✐t❡(UNIV/

/R)

❚❤❡r❡❢♦r❡ ✂♥✐t❡(UNIV/

/ ≈L(r))✳ ◗❡❞✳

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✺✴✶✽

❛ ❆♥t✐♠✐r♦✈ ✬✾✺

slide-46
SLIDE 46

What Have We Achieved?

✂♥✐t❡ (UNIV/

/ ≈A) ⇔ A ✐s r❡❣✉❧❛r

r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s ❛r❡ ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥❀ t❤✐s ✐s ♥♦✇ ❡❛s② ♥♦♥✲r❡❣✉❧❛r✐t② ✭ ✮ t❛❦❡ ❛♥② ❧❛♥❣✉❛❣❡❀ ❜✉✐❧❞ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜str✐♥❣s t❤❡♥ t❤✐s ❧❛♥❣✉❛❣❡ ✐s r❡❣✉❧❛r ✭ ✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✻✴✶✽

slide-47
SLIDE 47

What Have We Achieved?

✂♥✐t❡ (UNIV/

/ ≈A) ⇔ A ✐s r❡❣✉❧❛r

r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s ❛r❡ ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥❀ t❤✐s ✐s ♥♦✇ ❡❛s②

UNIV/ / ≈A = UNIV/ / ≈A

♥♦♥✲r❡❣✉❧❛r✐t② ✭ ✮ t❛❦❡ ❛♥② ❧❛♥❣✉❛❣❡❀ ❜✉✐❧❞ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜str✐♥❣s t❤❡♥ t❤✐s ❧❛♥❣✉❛❣❡ ✐s r❡❣✉❧❛r ✭ ✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✻✴✶✽

x ≈A y

❞❡❢

= ∀z. x@z ∈ A ⇔ y@z ∈ A

slide-48
SLIDE 48

What Have We Achieved?

✂♥✐t❡ (UNIV/

/ ≈A) ⇔ A ✐s r❡❣✉❧❛r

r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s ❛r❡ ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥❀ t❤✐s ✐s ♥♦✇ ❡❛s②

UNIV/ / ≈A = UNIV/ / ≈A

♥♦♥✲r❡❣✉❧❛r✐t② ✭anbn✮ t❛❦❡ ❛♥② ❧❛♥❣✉❛❣❡❀ ❜✉✐❧❞ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜str✐♥❣s t❤❡♥ t❤✐s ❧❛♥❣✉❛❣❡ ✐s r❡❣✉❧❛r ✭ ✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✻✴✶✽

slide-49
SLIDE 49

What Have We Achieved?

✂♥✐t❡ (UNIV/

/ ≈A) ⇔ A ✐s r❡❣✉❧❛r

r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s ❛r❡ ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥❀ t❤✐s ✐s ♥♦✇ ❡❛s②

UNIV/ / ≈A = UNIV/ / ≈A

♥♦♥✲r❡❣✉❧❛r✐t② ✭anbn✮ t❛❦❡ ❛♥② ❧❛♥❣✉❛❣❡❀ ❜✉✐❧❞ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜str✐♥❣s t❤❡♥ t❤✐s ❧❛♥❣✉❛❣❡ ✐s r❡❣✉❧❛r ✭ ✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✻✴✶✽

■❢ t❤❡r❡ ❡①✐sts ❛ s✉❢✂❝✐❡♥t❧② ❧❛r❣❡ s❡t B ✭❢♦r ❡①❛♠♣❧❡ ✐♥✂♥✐t❡❧② ❧❛r❣❡✮✱ s✉❝❤ t❤❛t

∀x, y ∈ B. x = y ⇒ x ≈A y✳

t❤❡♥ A ✐s ♥♦t r❡❣✉❧❛r✳

✭B

❞❡❢

=

n an✮

slide-50
SLIDE 50

What Have We Achieved?

✂♥✐t❡ (UNIV/

/ ≈A) ⇔ A ✐s r❡❣✉❧❛r

r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s ❛r❡ ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥❀ t❤✐s ✐s ♥♦✇ ❡❛s②

UNIV/ / ≈A = UNIV/ / ≈A

♥♦♥✲r❡❣✉❧❛r✐t② ✭anbn✮ t❛❦❡ ❛♥② ❧❛♥❣✉❛❣❡❀ ❜✉✐❧❞ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜str✐♥❣s t❤❡♥ t❤✐s ❧❛♥❣✉❛❣❡ ✐s r❡❣✉❧❛r ✭ ✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✻✴✶✽

slide-51
SLIDE 51

What Have We Achieved?

✂♥✐t❡ (UNIV/

/ ≈A) ⇔ A ✐s r❡❣✉❧❛r

r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s ❛r❡ ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥❀ t❤✐s ✐s ♥♦✇ ❡❛s②

UNIV/ / ≈A = UNIV/ / ≈A

♥♦♥✲r❡❣✉❧❛r✐t② ✭anbn✮ t❛❦❡ ❛♥② ❧❛♥❣✉❛❣❡❀ ❜✉✐❧❞ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ s✉❜str✐♥❣s t❤❡♥ t❤✐s ❧❛♥❣✉❛❣❡ ✐s r❡❣✉❧❛r ✭anbn ⇒ a⋆b⋆✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✻✴✶✽

slide-52
SLIDE 52

Conclusion

❲❡ ❤❛✈❡ ♥❡✈❡r s❡❡♥ ❛ ♣r♦♦❢ ♦❢ ▼②❤✐❧❧✲◆❡r♦❞❡ ❜❛s❡❞ ♦♥ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✳ ❣r❡❛t s♦✉r❝❡ ♦❢ ❡①❛♠♣❧❡s ✭✐♥❞✉❝t✐♦♥s✮ ♥♦ ♥❡❡❞ t♦ ✂❣❤t t❤❡ t❤❡♦r❡♠ ♣r♦✈❡r✿

✂rst ❞✐r❡❝t✐♦♥ ✭✼✾✵ ❧♦❝✮ s❡❝♦♥❞ ❞✐r❡❝t✐♦♥ ✭✹✵✵ ✴ ✸✾✵ ❧♦❝✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✼✴✶✽

slide-53
SLIDE 53

Conclusion

❲❡ ❤❛✈❡ ♥❡✈❡r s❡❡♥ ❛ ♣r♦♦❢ ♦❢ ▼②❤✐❧❧✲◆❡r♦❞❡ ❜❛s❡❞ ♦♥ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✳ ❣r❡❛t s♦✉r❝❡ ♦❢ ❡①❛♠♣❧❡s ✭✐♥❞✉❝t✐♦♥s✮ ♥♦ ♥❡❡❞ t♦ ✂❣❤t t❤❡ t❤❡♦r❡♠ ♣r♦✈❡r✿

✂rst ❞✐r❡❝t✐♦♥ ✭✼✾✵ ❧♦❝✮ s❡❝♦♥❞ ❞✐r❡❝t✐♦♥ ✭✹✵✵ ✴ ✸✾✵ ❧♦❝✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✼✴✶✽

slide-54
SLIDE 54

Conclusion

❲❡ ❤❛✈❡ ♥❡✈❡r s❡❡♥ ❛ ♣r♦♦❢ ♦❢ ▼②❤✐❧❧✲◆❡r♦❞❡ ❜❛s❡❞ ♦♥ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✳ ❣r❡❛t s♦✉r❝❡ ♦❢ ❡①❛♠♣❧❡s ✭✐♥❞✉❝t✐♦♥s✮ ♥♦ ♥❡❡❞ t♦ ✂❣❤t t❤❡ t❤❡♦r❡♠ ♣r♦✈❡r✿

✂rst ❞✐r❡❝t✐♦♥ ✭✼✾✵ ❧♦❝✮ s❡❝♦♥❞ ❞✐r❡❝t✐♦♥ ✭✹✵✵ ✴ ✸✾✵ ❧♦❝✮

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✼✴✶✽

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SLIDE 55

Thank you! Questions?

❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✶✽✴✶✽