By Thomas Baeck, D.B Fogel, Z Michalewicz
Quantity I supplied the final thought of evolutionary computation. This moment quantity nevertheless goals at introducing the reader to more effective features of evolutionary computation. whereas i discovered the 1st quantity nice, this moment quantity lacked the main points which are required to supply an instinct of the operating of complicated evolutionary thoughts. i believe that "How to resolve it" by means of Michalewicz and Fogel and "Genetic algorithms + info buildings = evolution courses" by way of Michalewicz either supply this adventure worthwhile to enforce evolutionary thoughts, through now not attempting to trade-off pages for understandability. i wouldn't suggest this ebook since it attempts to introduce complicated elements which are too tricky to hide in one bankruptcy every one. in the event you actually need to appreciate the perform of evolutionary concepts, you would like a great instinct of the way some of the operators and constructions paintings on genuine difficulties, simply analyzing a couple of pages won't do the task.
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T❻✂➁✏❽➝➉✺➄✬➄➆❻✂➁➞➨❜➁✏↔➒❼➂➊❢➊✂❼➂➊✂↔❯➎✺➟❑➄➆❻✂➁➞❿➂❼➂➊❢➁➣➙ ➔❩➉Ô➄➍➃t❻✂➁✏❽➝➉✺➄✬➄➆❻✂➁➞➁✫➊✂➐✦➎➣➟❑❿➂❼❾➊✂➁➣➙ • \( • \) • \| ❿❾➁✏➟ ➄❊➺❫➎➒→✸➁✫➊✂❼❾➊✂↔✴➼●➨✂➇➆➉➣➃t➫➣➁➧➄✤➙ ➇➏❼➂↔➒❻➌➄❊➺❫➃✫❿❾➎➒❽➆❼❾➊✂↔✴➼●➨✂➇➆➉➣➃t➫➣➁➧➄✤➙ ➄➆➇➍➉➣➊❢❽➆❿➮➉Ô➄➍➁✫❽✬❼❾➊➌➄➍➎ ∪ ➙ ❺ ❻✴➅✂❽✏➭ Ò ❲ + ✗ ➄➍➇➍➉✺➊✂❽➆❿➂➉✺➄➍➁✏❽●❼➂➊➌➄➆➎ ✬ ➙❐✒➜➎➣➄➍❼❾➃✫➁➜➄➍❻❪➉✺➄●❼➂➊ ➽❊➾ ➉➒➚➓➪✙➄➆❻✂➁✫➇➏➁ ( ∪ ) ? \(a\|b\)+d? x ➄➍➉➣➫➣➁✬➄➍➎✘❽➏➁✫➁❨➡❨❻✂➁✏➄➆❻✂➁✫➇✪➎➒➇✝➊❢➎➣➄ A ê✪✫✝➸❁❼❾➐✂➁✫➊➌➄➍❿➥➤➒0➭➌➄➍1❻❢➁➱➉✺➊✂❽➏n−1 ➡✍➁✫➇✪➐✂➁✏→❜➁✏➊✂➐✂❽✉➎➒➊➢➨❜➎➣➄➆❻ x ∈ L(A) ➉➣➊✂➐ ✏ ➙ ✒➱➎✺➄➍❼➂➃✏➁➜➄➆❻❪➉✺➄ ❼➥➟✩➉➣➊✂➐✧➎➒➊✂❿❾➤➢❼➥➟✼➄➆❻✂➁✫➇➏➁➀➉➣➇➏➁ ❽➏➅✂➃t❻✿➄➍❻❪➉Ô➄ A ➺ à★⑥ ✑➌➼ x x ∈ L(A) x x x qi , i < n + 1 xn−1 i0 = q0 →0 q1 →1 q2 →2 q2 .
Xj i0 −→ qj+1 ❼➥➬ qj+1 ∈ Hj+1 å➜➁✫➊✂➃✏➁➣➭ ❼❾➬ ➙✉ý❪➎➒➇✉➄➍❻✂➁✏➊✡➄➆❻✂➁✫➇➏➁✘➁➧➻➑❼➂❽↕➄➍❽❨➉➣➊✡➉✺➃✫➃✫➁✏→❢➄➍❼❾➊✂↔➢❽➏➄➍➉✺➄➍➁➀❼❾➊ x ∈ L(A) Hn ∩ F = ∅ ➙➀å➱➁✏➇➆➁❝➁✫➉➣➃t❻➋❽➏➄➆➁✫→☞➄➍➉➣➫➣➁✏❽➝➄➍❼❾➔✧➁❊➵✴➅❪➉➣➐✂➇➆➉✺➄➍❼❾➃➈❼❾➊➓➄➍❻✂➁➈➊✴➅✂➔❝➨✸➁✫➇➀➎➣➟❂❽➏➄t➉Ô➄➍➁✫❽✏ç➜➟❫➎➣➇➱↔➒❼➥➸➒➁✏➊ Hn ➭✂➡●➁➞➃✏➎➒➔✧→❢➅❢➄➍➁ ➨➌➤✖➐✂➎➣❼➂➊✂↔ ➔✧➉➣➊➌➤✿❿❾➎➑➎➣➫❁➅❢→✂❽⑦➟❫➎➒➇✬➁✏➸➣➁✫➇↕➤ ➙✪å➜➎❄➡✍➁✏➸➣➁✫➇✏➭ Hj Hj+1 Q q ∈ Hj ➄➆❻✂❼➂❽➈➊❁➅❢➔❝➨❜➁✏➇❯❼➂❽➈➨❪➉➣❽➆❼❾➃✤➉➣❿❾❿❾➤➩➨✸➎➒➅✂➊✂➐✂➁✏➐❅➟❫➎➒➇✛↔➒❼➥➸➒➁✏➊ ➙ ✰ ➎☞➄➆❻✂➁❩➄➍❼❾➔✧➁✧➇➆➁✏➵❁➅❢❼➂➇➆➁✏➔✧➁✏➊➌➄❯❼➂❽ A ➐✂➎✷➡❨➊✣➄➍➎➩➉➩➃✏➎➒➊✂❽↕➄t➉➣➊➌➄✧➐✂➁✫→✸➁✫➊✂➐❢❼➂➊✂↔➩➎➒➊ ➄➍❼❾➔✧➁✏❽❝➄➍❻✂➁✡❿➂➁✏➊✂↔➣➄➆❻Û➎➣➟ ➙✢❺✬❻✂❼➂❽☎❼➂❽❯➔✛➅✂➃t❻ A ➨✸➁✏➄➏➄➍➁✫➇✏➙✡å➱➎✷➡✍➁✏➸➒➁✏➇✫➭✝❼❾➊➳→✂➇➆➉➣➃✏➄➆❼➂➃✫➉➣❿✝➄➍➁✫➇➏➔✧❽➞ ➄➍❻❢❼➂❽❊❼➂❽➞❽➏➄➆❼➂❿➂❿✝➊❢➎➣➄✛↔➣➎➑➎❁➐➳➁✏x➊✂➎➒➅✂↔➣❻♥✬ ➙ ❞●➁✫➃✫➉➣➅✂❽➏➁ ➄➆❻✂➁❯➃✫➎➒➊❢❽➏➄t➉✺➊✴➄➞❼➥➄➝➄➍➉➣➫➣➁✏❽➞➄➆➎✖➃✫➎➒➔☎→✂➅❢➄➆➁❯➉✿❽➆❼❾➊✂↔➒❿❾➁✛❽➏➄➆➁✫→➩❼➂❽➝➄➆➎➑➎❩❿➮➉➣➇➏↔➒➁➣➙➞❺✬❻✂❼❾❽➀❼➂❽➀➨✸➁✫➃✫➉➣➅✂❽➏➁ ➄➍➎ ➙è➯➶➟➞➄➍❻✂➁➋❽↕➄➍➇➆❼❾➊✂↔➦❼❾❽❩➸➒➁✏➇➏➤❵❿❾➎➒➊✂↔✂➭⑦➄➆❻✂❼➂❽ ➡✍➁➋➇➆➁✏➃✫➎➒➔☎→✂➅❢➄➆➁➓➄➍❻✂➁☞➄➆➇➍➉➣➊❢❽➆❼❾➄➆❼➂➎➒➊ Hj Hj+1 ➔☎➁✤➉➣➊❢❽❨➄➆❻❪➉✺➄❨➡✍➁➈➇➏➁✫➃✫➎➣➔✧→✂➅➑➄➍➁➞➄➍❻✂➁➞❽➍➉✺➔✧➁➞→✂➇➏➎➒➨✂❿➂➁✏➔❶➎✷➸➒➁✏➇➀➉➣➊✂➐✦➎✷➸➒➁✏➇✫➙⑦➯➲➊✂❽↕➄➍➁✫➉➣➐♥➭❪➡●➁❊➃✤➉✺➊ →✂➇➏➁✫➃✏➎➒➔✧→❢➅❢➄➍➁❨➉➣❿❾❿➑➄➆❻✂➁⑦➄➍➇➆➉➣➊✂❽➆❼➥➄➍❼❾➎➒➊✂❽✫➙❑➯➶➄✝➄➍➅✂➇➏➊✂❽✝➎➒➅❢➄✝➄➆❻❪➉✺➄✝➡✍➁❨➃✤➉➣➊➢➐✂s➁ ✮❪➊✂➁➜➉➣➊☎➉➣➅❢➄➆➎➒➔❩➉Ô➄➍➎➒➊ ❼❾➊✡➄➆❻✂❼➂❽⑦➡●➉✤➤✖➄➍❻❪➉Ô➄✬➡●➁➞➃✤➉✺➊➠➅✂❽➏➁➈❼❾➊✿➄➍❻✂➁➞→✂❿➂➉➣➃✫➁➞➎➣➟ ➙ A ❀②❋➁❼➂s➃ A = A, Q, i , F, δ ➣✶⑤❩⑨✽➥➆➟➠➂s➃ Q : U ∩ F = ∅} ❷✇❸❛❹✚❺❼❻❾❽❫❻✕❿✽❺ ➺ à★⑥ ✳➒➼ ➄ 0 δ ℘ = { H, a, J : ➋ ➊❙➝✈⑦✘➟❾➟ q∈H ↔✻➉❶➂s⑨ ④③ ➂✇⑦✘⑨í⑦✘➢❱➃➻➊❙➞➆⑦✘➃❢➊❙⑨✽➣ ➈➢❱➃ ➃❾➉❶➂s➝➸➂ F ℘ := {U ⊆ ➅ a ➦ q ∈J q→q} ⑥ ➺ à★⑥ ❚➌➼ A℘ = A, ℘(Q), {i0 }, F ℘ , δ ℘ ➅✲➤➑⑦✘➟❾➟➠➂➑➥➆➃✢➉❶➂✼→ ❛➔❤↕❝➯✟→✁➯✓➒✞➏✞➍ ÿ↕ ➙ ➣ A ➦ ❆❈ ❀⑧⑨⑤ ✡⑦ ☎⑩ ❷❶ ❷✇❸❛❹✚❺❼❻❾❽❫❻✕❿✽❺ ❩⑨②⑦✘➢❱➃➻➊❙➞➆⑦✘➃➻➊❙⑨ ➅ ✻→✁➒➸→✁➳ ❨➏➻➯Ò➏➻➎✘➒❢➏➑➐ ➋❐➋ ➊❙➝➆⑦✘➟❾➟ ⑦✘⑨✽➥ ➦ ➦ q ∈Q a∈A ➃❾➉❶➂s➝➸➂ ➅✈⑦✘➃➈➞❑➊★➅s➃✿➊❙⑨❜➂ ➅s➢➹➤➡➉✪➃❾➉♥⑦✘➃ ➣ a ➦ q ∈Q q→q à➒à ➙ ➾ ➎➒➔☎→✂❿➂➁þ➻❢❼➥➄➲➤✖➉➣➊❢➐➓➚✦❼➂➊✂❼❾➔❩➉➣❿❜➛➝➅❢➄➆➎➒➔✧➉✺➄t➉ ✜ ⑥ ➯➶➄✛❼➂❽❊➃✫❿➂➁✫➉➣➇➈➄➆❻❪➉✺➄❊➟❫➎➒➇❝➉✦➐✂➁✏➄➆➁✫➇➏➔✧❼❾➊✂❼➂❽↕➄➍❼➂➃☎➉➣➊❢➐ ➄➍➎➣➄➍➉➣❿✍➉➣➅❢➄➍➎➣➔❩➉✺➄➆➎➒➊♥➭❑➉➣❿➂❿✝➡✍➁✿❻❪➉✤➸➒➁✧➄➍➎☞➐✂➎ ❼❾❽➝➄➍➎✿❿❾➎➑➎➣➫✦➅❢→✙➄➆❻✂➁❯➊✂➁þ➻➑➄➞❽↕➄t➉✺➄➆➁❝❼➂➊✣➺ ★à ⑥ ✑➌➼t➭✼➡❨❻✂❼❾➃t❻✙➁➧➻➑❼➂❽↕➄➍❽➞➉➣➊✂➐➋❼❾❽➀➅✂➊✂❼❾➵✴➅✂➁➣➙➈ý❪➎➒➇➱➄➆❻✂➁✫❽➏➁ ➉➣➅➑➄➍➎➒➔✧➉✺➄t➉➑➭❢➇➆➁✏➃✫➎➒↔➣➊✂➠❼ ô✏❼➂➊✂↔❝➄➆❻✂➁➞❿➮➉➣➊✂↔➣➅❪➉➣↔➒➁➀❼❾❽✬❿➂❼❾➊✂➁✤➉➣➇●❼➂➊✖➄➍❻✂➁➞❽↕➄➍➇➆❼❾➊✂↔✂➙ ▲☛ ❹❸❺♠ ❝❻ ❯❋ ✑✏ ã ❸✡❿➹Ï❫❸ ❼ ➀ ❼➊❙➝✇➂s➜✘➂s➝⑩❂⑦✘➢❱➃➻➊❙➞➆⑦✘➃❢➊❙⑨ ➇☞➃❾➉❶➂✬➂ ❃➩❶➊❙⑨❜➂s⑨❶➃ ⑦✘➟ ℘ ➅➫➃❢➊❙➃ó⑦✘➟ ⑦✘⑨✽➥Û➥✛➂ A ➦ A ➦ ➣ ➃➻➂s➝➞ ⑨ ➅s➃ ➤➛➣ ➧➊❙➝➸➂➡➊❙➜✘➂s➝➸➇ ℘ L(A ) = L(A) ➦ ➦ ➦ ✜✣ ✰ ➎❢➭♥➄➍❻✂➁❯➇➏➁✫➃✫❼❾→❜➁❝➄➆➎✡➉✺➄➆➄➍➉➣➃t➫➓➄➆❻✂➁❯→✂➇➆➎➣➨✂❿➂➁✏➔ ê ❼➂❽➱➄➍❻✂❼❾❽✫ç❩✮❪➇➏❽➏➄➞➃✏➎➒➔✧→❢➅❢➄➍➁ ℘ x ∈ L(A) ➉➣➊❢➐✛➄➍❻✂➁✏➊❝➃t❻✂➁✏➃t➫ ➙ ✰ ❼➂➊✂➃✏➁✉➄➆❻✂➁●❿➂➉✺➄➆➄➆➁✫➇❏❼➂❽❏➐✂➁✏➄➆➁✫➇➏➔✧❼❾➊✂❼➂❽↕➄➍❼➂➃✺➭Ô➄➆❻✂➁✍➄➆❼➂➔☎➁✪➊✂➁✫➁✏➐✂A➁✫➐ ❼❾❽➈➉✺➃✏➄➍➅✂➉➣❿➂❿➥➤☞❿➂❼❾➊✂➁✤➉✺x➇✘❼➂∈➊✙L(A)?
Xj i0 −→ qj+1 ❼➥➬ qj+1 ∈ Hj+1 å➜➁✫➊✂➃✏➁➣➭ ❼❾➬ ➙✉ý❪➎➒➇✉➄➍❻✂➁✏➊✡➄➆❻✂➁✫➇➏➁✘➁➧➻➑❼➂❽↕➄➍❽❨➉➣➊✡➉✺➃✫➃✫➁✏→❢➄➍❼❾➊✂↔➢❽➏➄➍➉✺➄➍➁➀❼❾➊ x ∈ L(A) Hn ∩ F = ∅ ➙➀å➱➁✏➇➆➁❝➁✫➉➣➃t❻➋❽➏➄➆➁✫→☞➄➍➉➣➫➣➁✏❽➝➄➍❼❾➔✧➁❊➵✴➅❪➉➣➐✂➇➆➉✺➄➍❼❾➃➈❼❾➊➓➄➍❻✂➁➈➊✴➅✂➔❝➨✸➁✫➇➀➎➣➟❂❽➏➄t➉Ô➄➍➁✫❽✏ç➜➟❫➎➣➇➱↔➒❼➥➸➒➁✏➊ Hn ➭✂➡●➁➞➃✏➎➒➔✧→❢➅❢➄➍➁ ➨➌➤✖➐✂➎➣❼➂➊✂↔ ➔✧➉➣➊➌➤✿❿❾➎➑➎➣➫❁➅❢→✂❽⑦➟❫➎➒➇✬➁✏➸➣➁✫➇↕➤ ➙✪å➜➎❄➡✍➁✏➸➣➁✫➇✏➭ Hj Hj+1 Q q ∈ Hj ➄➆❻✂❼➂❽➈➊❁➅❢➔❝➨❜➁✏➇❯❼➂❽➈➨❪➉➣❽➆❼❾➃✤➉➣❿❾❿❾➤➩➨✸➎➒➅✂➊✂➐✂➁✏➐❅➟❫➎➒➇✛↔➒❼➥➸➒➁✏➊ ➙ ✰ ➎☞➄➆❻✂➁❩➄➍❼❾➔✧➁✧➇➆➁✏➵❁➅❢❼➂➇➆➁✏➔✧➁✏➊➌➄❯❼➂❽ A ➐✂➎✷➡❨➊✣➄➍➎➩➉➩➃✏➎➒➊✂❽↕➄t➉➣➊➌➄✧➐✂➁✫→✸➁✫➊✂➐❢❼➂➊✂↔➩➎➒➊ ➄➍❼❾➔✧➁✏❽❝➄➍❻✂➁✡❿➂➁✏➊✂↔➣➄➆❻Û➎➣➟ ➙✢❺✬❻✂❼➂❽☎❼➂❽❯➔✛➅✂➃t❻ A ➨✸➁✏➄➏➄➍➁✫➇✏➙✡å➱➎✷➡✍➁✏➸➒➁✏➇✫➭✝❼❾➊➳→✂➇➆➉➣➃✏➄➆❼➂➃✫➉➣❿✝➄➍➁✫➇➏➔✧❽➞ ➄➍❻❢❼➂❽❊❼➂❽➞❽➏➄➆❼➂❿➂❿✝➊❢➎➣➄✛↔➣➎➑➎❁➐➳➁✏x➊✂➎➒➅✂↔➣❻♥✬ ➙ ❞●➁✫➃✫➉➣➅✂❽➏➁ ➄➆❻✂➁❯➃✫➎➒➊❢❽➏➄t➉✺➊✴➄➞❼➥➄➝➄➍➉➣➫➣➁✏❽➞➄➆➎✖➃✫➎➒➔☎→✂➅❢➄➆➁❯➉✿❽➆❼❾➊✂↔➒❿❾➁✛❽➏➄➆➁✫→➩❼➂❽➝➄➆➎➑➎❩❿➮➉➣➇➏↔➒➁➣➙➞❺✬❻✂❼❾❽➀❼➂❽➀➨✸➁✫➃✫➉➣➅✂❽➏➁ ➄➍➎ ➙è➯➶➟➞➄➍❻✂➁➋❽↕➄➍➇➆❼❾➊✂↔➦❼❾❽❩➸➒➁✏➇➏➤❵❿❾➎➒➊✂↔✂➭⑦➄➆❻✂❼➂❽ ➡✍➁➋➇➆➁✏➃✫➎➒➔☎→✂➅❢➄➆➁➓➄➍❻✂➁☞➄➆➇➍➉➣➊❢❽➆❼❾➄➆❼➂➎➒➊ Hj Hj+1 ➔☎➁✤➉➣➊❢❽❨➄➆❻❪➉✺➄❨➡✍➁➈➇➏➁✫➃✫➎➣➔✧→✂➅➑➄➍➁➞➄➍❻✂➁➞❽➍➉✺➔✧➁➞→✂➇➏➎➒➨✂❿➂➁✏➔❶➎✷➸➒➁✏➇➀➉➣➊✂➐✦➎✷➸➒➁✏➇✫➙⑦➯➲➊✂❽↕➄➍➁✫➉➣➐♥➭❪➡●➁❊➃✤➉✺➊ →✂➇➏➁✫➃✏➎➒➔✧→❢➅❢➄➍➁❨➉➣❿❾❿➑➄➆❻✂➁⑦➄➍➇➆➉➣➊✂❽➆❼➥➄➍❼❾➎➒➊✂❽✫➙❑➯➶➄✝➄➍➅✂➇➏➊✂❽✝➎➒➅❢➄✝➄➆❻❪➉✺➄✝➡✍➁❨➃✤➉➣➊➢➐✂s➁ ✮❪➊✂➁➜➉➣➊☎➉➣➅❢➄➆➎➒➔❩➉Ô➄➍➎➒➊ ❼❾➊✡➄➆❻✂❼➂❽⑦➡●➉✤➤✖➄➍❻❪➉Ô➄✬➡●➁➞➃✤➉✺➊➠➅✂❽➏➁➈❼❾➊✿➄➍❻✂➁➞→✂❿➂➉➣➃✫➁➞➎➣➟ ➙ A ❀②❋➁❼➂s➃ A = A, Q, i , F, δ ➣✶⑤❩⑨✽➥➆➟➠➂s➃ Q : U ∩ F = ∅} ❷✇❸❛❹✚❺❼❻❾❽❫❻✕❿✽❺ ➺ à★⑥ ✳➒➼ ➄ 0 δ ℘ = { H, a, J : ➋ ➊❙➝✈⑦✘➟❾➟ q∈H ↔✻➉❶➂s⑨ ④③ ➂✇⑦✘⑨í⑦✘➢❱➃➻➊❙➞➆⑦✘➃❢➊❙⑨✽➣ ➈➢❱➃ ➃❾➉❶➂s➝➸➂ F ℘ := {U ⊆ ➅ a ➦ q ∈J q→q} ⑥ ➺ à★⑥ ❚➌➼ A℘ = A, ℘(Q), {i0 }, F ℘ , δ ℘ ➅✲➤➑⑦✘➟❾➟➠➂➑➥➆➃✢➉❶➂✼→ ❛➔❤↕❝➯✟→✁➯✓➒✞➏✞➍ ÿ↕ ➙ ➣ A ➦ ❆❈ ❀⑧⑨⑤ ✡⑦ ☎⑩ ❷❶ ❷✇❸❛❹✚❺❼❻❾❽❫❻✕❿✽❺ ❩⑨②⑦✘➢❱➃➻➊❙➞➆⑦✘➃➻➊❙⑨ ➅ ✻→✁➒➸→✁➳ ❨➏➻➯Ò➏➻➎✘➒❢➏➑➐ ➋❐➋ ➊❙➝➆⑦✘➟❾➟ ⑦✘⑨✽➥ ➦ ➦ q ∈Q a∈A ➃❾➉❶➂s➝➸➂ ➅✈⑦✘➃➈➞❑➊★➅s➃✿➊❙⑨❜➂ ➅s➢➹➤➡➉✪➃❾➉♥⑦✘➃ ➣ a ➦ q ∈Q q→q à➒à ➙ ➾ ➎➒➔☎→✂❿➂➁þ➻❢❼➥➄➲➤✖➉➣➊❢➐➓➚✦❼➂➊✂❼❾➔❩➉➣❿❜➛➝➅❢➄➆➎➒➔✧➉✺➄t➉ ✜ ⑥ ➯➶➄✛❼➂❽❊➃✫❿➂➁✫➉➣➇➈➄➆❻❪➉✺➄❊➟❫➎➒➇❝➉✦➐✂➁✏➄➆➁✫➇➏➔✧❼❾➊✂❼➂❽↕➄➍❼➂➃☎➉➣➊❢➐ ➄➍➎➣➄➍➉➣❿✍➉➣➅❢➄➍➎➣➔❩➉✺➄➆➎➒➊♥➭❑➉➣❿➂❿✝➡✍➁✿❻❪➉✤➸➒➁✧➄➍➎☞➐✂➎ ❼❾❽➝➄➍➎✿❿❾➎➑➎➣➫✦➅❢→✙➄➆❻✂➁❯➊✂➁þ➻➑➄➞❽↕➄t➉✺➄➆➁❝❼➂➊✣➺ ★à ⑥ ✑➌➼t➭✼➡❨❻✂❼❾➃t❻✙➁➧➻➑❼➂❽↕➄➍❽➞➉➣➊✂➐➋❼❾❽➀➅✂➊✂❼❾➵✴➅✂➁➣➙➈ý❪➎➒➇➱➄➆❻✂➁✫❽➏➁ ➉➣➅➑➄➍➎➒➔✧➉✺➄t➉➑➭❢➇➆➁✏➃✫➎➒↔➣➊✂➠❼ ô✏❼➂➊✂↔❝➄➆❻✂➁➞❿➮➉➣➊✂↔➣➅❪➉➣↔➒➁➀❼❾❽✬❿➂❼❾➊✂➁✤➉➣➇●❼➂➊✖➄➍❻✂➁➞❽↕➄➍➇➆❼❾➊✂↔✂➙ ▲☛ ❹❸❺♠ ❝❻ ❯❋ ✑✏ ã ❸✡❿➹Ï❫❸ ❼ ➀ ❼➊❙➝✇➂s➜✘➂s➝⑩❂⑦✘➢❱➃➻➊❙➞➆⑦✘➃❢➊❙⑨ ➇☞➃❾➉❶➂✬➂ ❃➩❶➊❙⑨❜➂s⑨❶➃ ⑦✘➟ ℘ ➅➫➃❢➊❙➃ó⑦✘➟ ⑦✘⑨✽➥Û➥✛➂ A ➦ A ➦ ➣ ➃➻➂s➝➞ ⑨ ➅s➃ ➤➛➣ ➧➊❙➝➸➂➡➊❙➜✘➂s➝➸➇ ℘ L(A ) = L(A) ➦ ➦ ➦ ✜✣ ✰ ➎❢➭♥➄➍❻✂➁❯➇➏➁✫➃✫❼❾→❜➁❝➄➆➎✡➉✺➄➆➄➍➉➣➃t➫➓➄➆❻✂➁❯→✂➇➆➎➣➨✂❿➂➁✏➔ ê ❼➂❽➱➄➍❻✂❼❾❽✫ç❩✮❪➇➏❽➏➄➞➃✏➎➒➔✧→❢➅❢➄➍➁ ℘ x ∈ L(A) ➉➣➊❢➐✛➄➍❻✂➁✏➊❝➃t❻✂➁✏➃t➫ ➙ ✰ ❼➂➊✂➃✏➁✉➄➆❻✂➁●❿➂➉✺➄➆➄➆➁✫➇❏❼➂❽❏➐✂➁✏➄➆➁✫➇➏➔✧❼❾➊✂❼➂❽↕➄➍❼➂➃✺➭Ô➄➆❻✂➁✍➄➆❼➂➔☎➁✪➊✂➁✫➁✏➐✂A➁✫➐ ❼❾❽➈➉✺➃✏➄➍➅✂➉➣❿➂❿➥➤☞❿➂❼❾➊✂➁✤➉✺x➇✘❼➂∈➊✙L(A)?