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

  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 ♣✳ ✶✴✶✽

  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 ♣✳ ✶✴✶✽

  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 ♣✳ ✷✴✶✽

  4. r❡❧✐❡❞ ♦♥ t❤❡✐r ♣r♦♦❢ ✐♥ ❛ s❡❝✉r✐t② ❝r✐t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥ ❆♥❞r❡✇ ❆♣♣❡❧ ✭Pr✐♥❝❡t♦♥✮ ✭■ ❛❧s♦ ❢♦✉♥❞ ❛♥ ❡rr♦r ✐♥ ♠② P❤✳❉✳✲t❤❡s✐s ❛❜♦✉t ❝✉t✲❡❧✐♠✐♥❛t✐♦♥ ❡①❛♠✐♥❡❞ ❜② ❍❡♥❦ ❇❛r❡♥❞r❡❣t ❛♥❞ ❆♥❞② P✐tts✳✮ ♣✉❜❧✐s❤❡❞ ❛ ♣r♦♦❢ ✐♥ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲♦❣✐❝ ✱ ✷✵✵✺✱ ❋r❛♥❦ P❢❡♥♥✐♥❣ ∼ ✸✶♣♣ ❇♦❜ ❍❛r♣❡r ✭❈▼❯✮ ✭❈▼❯✮ ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✸✴✶✽

  5. ✭■ ❛❧s♦ ❢♦✉♥❞ ❛♥ ❡rr♦r ✐♥ ♠② P❤✳❉✳✲t❤❡s✐s ❛❜♦✉t ❝✉t✲❡❧✐♠✐♥❛t✐♦♥ ❡①❛♠✐♥❡❞ ❜② ❍❡♥❦ ❇❛r❡♥❞r❡❣t ❛♥❞ ❆♥❞② P✐tts✳✮ ♣✉❜❧✐s❤❡❞ ❛ ♣r♦♦❢ ✐♥ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲♦❣✐❝ ✱ ✷✵✵✺✱ ❋r❛♥❦ P❢❡♥♥✐♥❣ ∼ ✸✶♣♣ ❇♦❜ ❍❛r♣❡r ✭❈▼❯✮ ✭❈▼❯✮ r❡❧✐❡❞ ♦♥ t❤❡✐r ♣r♦♦❢ ✐♥ ❛ s❡❝✉r✐t② ❝r✐t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥ ❆♥❞r❡✇ ❆♣♣❡❧ ✭Pr✐♥❝❡t♦♥✮ ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✸✴✶✽

  6. ♣✉❜❧✐s❤❡❞ ❛ ♣r♦♦❢ ✐♥ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲♦❣✐❝ ✱ ✷✵✵✺✱ ❋r❛♥❦ P❢❡♥♥✐♥❣ ∼ ✸✶♣♣ ❇♦❜ ❍❛r♣❡r ✭❈▼❯✮ ✭❈▼❯✮ r❡❧✐❡❞ ♦♥ t❤❡✐r ♣r♦♦❢ ✐♥ ❛ s❡❝✉r✐t② ❝r✐t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥ ❆♥❞r❡✇ ❆♣♣❡❧ ✭Pr✐♥❝❡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 ♣✳ ✸✴✶✽

  7. ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s Formal language theory.. . ✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs ❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳ ❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

  8. Formal language theory.. . ✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs ❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳ ❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s A 1 A 2 ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

  9. Formal language theory.. . ✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs ❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳ ❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s ⇒ A 1 A 2 A 1 A 2 ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

  10. Formal language theory.. . ✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs ❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳ ❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s ⇒ A 1 A 2 A 1 A 2 ❞✐s❥♦✐♥t ✉♥✐♦♥✿ ❞❡❢ A 1 ⊎ A 2 = { (1 , x ) | x ∈ A 1 } ∪ { (2 , y ) | y ∈ A 2 } ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

  11. Formal language theory.. . ✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs ❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳ ❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s Pr♦❜❧❡♠s ✇✐t❤ ❞❡✂♥✐t✐♦♥ ❢♦r r❡❣✉❧❛r✐t②✿ ⇒ A 1 A 2 A 1 A 2 ❞❡❢ ✐s❴r❡❣✉❧❛r ( A ) = ∃ M. ✐s❴❞❢❛ ( M ) ∧ L ( M ) = A ❞✐s❥♦✐♥t ✉♥✐♦♥✿ ❞❡❢ A 1 ⊎ A 2 = { (1 , x ) | x ∈ A 1 } ∪ { (2 , y ) | y ∈ A 2 } ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

  12. Formal language theory.. . ✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs ❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳ ❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s ⇒ A 1 A 2 A 1 A 2 ❆ s♦❧✉t✐♦♥✿ ✉s❡ ♥❛ts ⇒ st❛t❡ ♥♦❞❡s ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

  13. Formal language theory.. . ✐♥ ❚❤❡♦r❡♠ Pr♦✈❡rs ❡✳❣✳ ■s❛❜❡❧❧❡✱ ❈♦q✱ ❍❖▲✹✱ ✳ ✳ ✳ ❛✉t♦♠❛t❛ ⇒ ❣r❛♣❤s✱ ♠❛tr✐❝❡s✱ ❢✉♥❝t✐♦♥s ❝♦♠❜✐♥✐♥❣ ❛✉t♦♠❛t❛✴❣r❛♣❤s ⇒ A 1 A 2 A 1 A 2 ❆ s♦❧✉t✐♦♥✿ ✉s❡ ♥❛ts ⇒ st❛t❡ ♥♦❞❡s ❨♦✉ ❤❛✈❡ t♦ r❡♥❛♠❡ st❛t❡s✦ ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✹✴✶✽

  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 ♣✳ ✺✴✶✽

  15. ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛ ✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛ ■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A ✳ ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

  16. ♠♦st t❡①t❜♦♦❦s ❛r❡ ❛❜♦✉t ❛✉t♦♠❛t❛ ■♥❢r❛str✉❝t✉r❡ ❢♦r ❢r❡❡✳ ❇✉t ❞♦ ✇❡ ❧♦s❡ ❛♥②t❤✐♥❣❄ ♣✉♠♣✐♥❣ ❧❡♠♠❛ ❝❧♦s✉r❡ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ❉❡✂♥✐t✐♦♥✿ ❆ ❧❛♥❣✉❛❣❡ A ✐s r❡❣✉❧❛r✱ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ t❤❛t ♠❛t❝❤❡s ❛❧❧ str✐♥❣s ♦❢ A ✳ ✳ ✳ ✳ ❛♥❞ ❢♦r❣❡t ❛❜♦✉t ❛✉t♦♠❛t❛ ❙t ❆♥❞r❡✇s✱ ✶✾ ◆♦✈❡♠❜❡r ✷✵✶✶ ➊ ♣✳ ✻✴✶✽

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