Otomat hữu hạn và ngôn ngữ chính quy (2018)

TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* LÊ MINH PHƯƠNG OTOMAT HỮU HẠN VÀ NGÔN NGỮ CHÍNH QUY KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chun ngành: Toán ứng dụng HÀ NỘI – 2018 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* LÊ MINH PHƯƠNG OTOMAT HỮU HẠN VÀ NGƠN NGỮ CHÍNH QUY KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Toán ứng dụng Người hướng dẫn khoa học TS KIỀU VĂN HƯNG HÀ NỘI – 2018 ▲❮■ ❈❷▼ ❒◆ ❊♠ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ tê ❚♦→♥ ù♥❣ ❞ư♥❣✱ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ✤➣ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ❑✐➲✉ ❱➠♥ ❍÷♥❣ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✱ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ❝â ♥❤ú♥❣ t❤✐➳✉ sât ✈➔ ❤↕♥ ❝❤➳✳ ❑➼♥❤ ữủ sỹ õ õ ỵ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ ✈➔ t♦➔♥ t❤➸ ❜↕♥ ✤å❝ ✤➸ ✤➲ t➔✐ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ ♥❣➔② ✷✵ t❤→♥❣ ✵✹ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ▲❮■ ❈❆▼ ✣❖❆◆ ❊♠ ①✐♥ ữợ sỹ ữợ t ❍÷♥❣ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❦❤ỉ♥❣ trò♥❣ ✈ỵ✐ ❜➜t ❦➻ ✤➲ t➔✐ ♥➔♦ ❦❤→❝✳ ❚r♦♥❣ ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❡♠ ✤➣ sû ❞ö♥❣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ ❧á♥❣ ❜✐➳t ì♥ ✈➔ tr➙♥ trå♥❣✳ ❍➔ ◆ë✐✱ ♥❣➔② ✷✵ t❤→♥❣ ✵✹ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ❱✐➯♥ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ▼ư❝ ❧ư❝ ▲í✐ ♠ð ✤➛✉ ✶ ✶ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ✷ ✶✳✶ ❇↔♥❣ ❝❤ú ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❚ø ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ◆❣æ♥ ♥❣ú ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹ ❱➠♥ ♣❤↕♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷ ❖❚❖▼❆❚ ❍Ú❯ ❍❸◆ ✶✹ ✷✳✶ ❚ê♥❣ q✉❛♥ ✈➲ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ❖t♦♠❛t ❤ú✉ ❤↕♥ ✤ì♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ❖t♦♠❛t ❤ú✉ ❤↕♥ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹ ❙ü t÷ì♥❣ ✤÷ì♥❣ ❣✐ú❛ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤ì♥ ✤à♥❤ ✈➔ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✺ ▼ët ù♥❣ ❞ö♥❣ ❝õ❛ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸ ◆●➷◆ ◆●Ú ❈❍➑◆❍ ◗❯❨ ✸✻ ✸✳✶ ❇✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷ ◆❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✸ ❙ü ❧✐➯♥ ❤➺ ❣✐ú❛ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✈➔ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② ✸✽ ✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ❑➳t ❧✉➟♥ ✹✹ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✹✺ ✐✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ▲í✐ ♠ð ✤➛✉ ◆❣æ♥ ♥❣ú ❤➻♥❤ t❤ù❝✱ ♠ët sü tê♥❣ q✉→t ❝õ❛ ổ ỳ tỹ ữợ ổ t ỗ tứ r tứ ữủ ♥❣❤✐➯♥ ❝ù✉ s➙✉ s➢❝ ❜ð✐ ◆✳ ❈❤♦♠s❦② ❝ò♥❣ ♥❤✐➲✉ t→❝ ỳ t trữợ ❚r♦♥❣ ✤â✱ ❧ỵ♣ ♥❣ỉ♥ ♥❣ú ❝â ♥❤✐➲✉ t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t ✈➔ ✤÷đ❝ ù♥❣ ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❧➽♥❤ ✈ü❝ ❝ỉ♥❣ ♥❣❤➺ t❤ỉ♥❣ t✐♥ ❧➔ ❧ỵ♣ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉②✳ ❑❤â❛ ❧✉➟♥ ♥➔② t➻♠ ❤✐➸✉ ✈➲ ❧ỵ♣ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉② ✈➔ ♠ët ❝ỉ♥❣ ❝ư ✤➸ ✤♦→♥ ♥❤➟♥ ♥â✱ ♦t♦♠❛t ❤ú✉ õ ỗ ữỡ ữỡ tự ❝ì sð✧ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥ ✈➲ ♥❣ỉ♥ ♥❣ú ❤➻♥❤ t❤ù❝ ♥❤÷ ❜↔♥❣ ❝❤ú✱ tø✱ ✈➠♥ ♣❤↕♠✳ ❈❤÷ì♥❣ ✷ ✧❖t♦♠❛t ❤ú✉ ❤↕♥✧ tr➻♥❤ ❜➔② ✈➲ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✲ ♠ët ♠æ ❤➻♥❤ ✧♠→② trứ tữủ ổ ỳ q ợ ❧♦↕✐ ♦t♦♠❛t ✤ì♥ ✤à♥❤ ✈➔ ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤✳ ❈❤÷ì♥❣ ✸ ✧◆❣æ♥ ♥❣ú ❝❤➼♥❤ q✉②✧ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➲ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✈➔ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉②✳ ✶ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➔ ❦➳t q✉↔ ✈➲ ♥❣ỉ♥ ♥❣ú ❤➻♥❤ t❤ù❝ s➩ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❝→❝ ❝❤÷ì♥❣ s❛✉✳ ❈→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❧➼ t❤✉②➳t t➟♣ ❤đ♣ ✈➔ ✤↕✐ sè ✤÷đ❝ ①❡♠ ♥❤÷ ❧➔ ✤➣ ❜✐➳t✳ ✶✳✶ ❇↔♥❣ ỳ rộ ỗ ỳ ỵ ữủ ỳ ộ ♣❤➛♥ tû a ∈ Σ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝❤ú ởt ỵ ♠ët sè ❜↔♥❣ ❝❤ú✿ ✰ Σ = {a, b, c, d, e, , z}❀ ✰ δ = {α, β, γ, δ}❀ ✰ Γ = {0, 1}✳ ✶✳✷ ❚ø ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❈❤♦ ❜↔♥❣ ❝❤ú Σ✱ ♠ët ❞➣② ❤ú✉ ❤↕♥ ❝→❝ ❝❤ú α = a1 a2 am ✈ỵ✐ ∈ Σ (0 ≤ i ≤ m) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✷ tø ❤❛② ①➙✉ tr➯♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ❜↔♥❣ ❝❤ú Σ✳ ❚ø ❦❤ỉ♥❣ ❝â ❝❤ú ♥➔♦ ✤÷đ❝ ❣å✐ ❧➔ tø ré♥❣ ✈➔ ❦➼ ❤✐➺✉ ❧➔ ε✳ ◆❤÷ ✈➟②✱ ♠ët tø tr➯♥ ❜↔♥❣ ❝❤ú Σ ❧➔ ởt ỳ ỗ ởt số ợ ỡ ❜➡♥❣ ❦❤æ♥❣ ❝→❝ ❝❤ú ❝→✐ ❝õ❛ Σ✱ tr♦♥❣ ✤â ♠ët ❝❤ú ❝→✐ ❝â t❤➸ ①✉➜t ❤✐➺♥ ♥❤✐➲✉ ❧➛♥✳ ❚ê♥❣ sè ✈à tr➼ ❝õ❛ ❝→❝ ❦➼ ❤✐➺✉ ①✉➜t ❤✐➺♥ tr♦♥❣ tø α ✤÷đ❝ ❣å✐ ❧➔ ✤ë ❞➔✐ ❝õ❛ tø α ✈➔ || ữợ || = tt ❝↔ ❝→❝ tø tr➯♥ ❜↔♥❣ ❝❤ú Σ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ Σ∗ ✈➔ Σ+ = Σ∗ \ {ε}✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ Σ∗ = Σ+ ∪ {ε}✳ ❱➼ ❞ö ✶✳✷✳ ❈❤♦ Σ = {a, b, c, , z}✱ Γ = {0, 1} ❧➔ ✷ ❜↔♥❣ ❝❤ú✳ ❑❤✐ ✤â ε✱ ❞❛❞✱ ♠♦♠✱ ❢♦♦❞ ❧➔ ❝→❝ tø tr➯♥ ❜↔♥❣ ❝❤ú Σ ✈➔ |dad| = |mom| = 3✱ |f ood| = 4✳ ε✱ ✶✱ ✶✵✱ ✶✵✵✶✱ ✶✵✶✶✵✵ ❧➔ ❝→❝ tø tr➯♥ ❜↔♥❣ ❝❤ú Γ ✈➔ |1| = 1✱ |10| = 2✱ |1001| = 4✱ |101100| = 6✳ ◆❤➟♥ ①➨t✿ ◆➳✉ α ❧➔ ♠ët tø tr➯♥ ❜↔♥❣ ❝❤ú Σ ✈➔ Σ ⊆ ∆ t❤➻ α ❝ô♥❣ ❧➔ tø tr➯♥ ❜↔♥❣ ❝❤ú ∆✳ ❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ tø✿ ❈❤♦ ❜↔♥❣ ❝❤ú Σ ✈➔ ❝→❝ tø α = a1 a2 am ✱ β = b1 b2 bn ✈➔ γ = c1 c2 ck (m, n, k ∈ N)✳ ❚❛ ①→❝ ✤à♥❤ ❝→❝ ♣❤➨♣ t♦→♥ s❛✉✿ ✲ P❤➨♣ ♥❤➙♥ ❣❤➨♣✿ ❚➼❝❤ ❣❤➨♣ ✭❤❛② ♥❤➙♥ ❣❤➨♣✮ ❝õ❛ ❤❛✐ tø α ✈➔ β ❧➔ tø γ = α.β = β.α = a1 a2 am b1 b2 bn ✳ ✲ P❤➨♣ ❧➜② tø ♥❣÷đ❝✿ ●✐↔ sû tø α = ε t❤➻ tø αR = amam−1 a2a1 ❧➔ tø ♥❣÷đ❝ ✭❤❛② tø s♦✐ ❣÷ì♥❣✮ ❝õ❛ tø α✳ ✲ P❤➨♣ ❝❤✐❛ tø✿ ❧➔ ♣❤➨♣ ♥❣➢t ❜ä ♣❤➛♥ ✤➛✉ ❤❛② ♣❤➛♥ ❝✉è✐ ❝õ❛ ♠ët tø✳ P❤➨♣ ❝❤✐❛ tr→✐ ✭♣❤↔✐✮ ❝õ❛ tø α ❝❤♦ tø β ✭γ ✮ ❤❛② ❣å✐ ❧➔ t❤÷ì♥❣ ❜➯♥ tr→✐ ✭♣❤↔✐✮ ❝õ❛ α ✈➔ β ✭γ ✮ ❝❤♦ ❦➳t q✉↔ ❧➔ ♣❤➛♥ ❝á♥ ❧↕✐ ❝õ❛ tø α ✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ s❛✉ ❦❤✐ ♥❣➢t ❜ä ♣❤➛♥ ✤➛✉ ✭❝✉è✐✮ β ✭γ ✮ tr♦♥❣ tø α✱ ✈➔ ữủ ỵ / \ ❞ö ✶✳✸✳ ❈❤♦ ❝→❝ tø α = abcaccabbc✱ β = abc✱ γ = bbc tr➯♥ ❜↔♥❣ ❝❤ú Σ = {a, b, c}✳ ❑❤✐ ✤â✿ ✰ α.β = abcaccabbcabc✱ γ.α = bbcabcaccabbc❀ ✰ αR = cbbaccacba✱ β R = cba❀ ✰ α /γ = abcacca ✱ β \α = accabbc✳ ✶✳✸ ◆❣æ♥ ♥❣ú ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❈❤♦ ❜↔♥❣ ❝❤ú Σ✱ ♠é✐ t➟♣ ❝♦♥ L ⊆ Σ∗ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥❣ỉ♥ ỳ tr ỳ ú ỵ r t rộ ởt ổ ỳ ổ ỗ ởt ❝❤ú ❝→✐ ♥➔♦ ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ♥❣ỉ♥ ♥❣ú ré♥❣✳ ◆❣æ♥ ♥❣ú ré♥❣ ❧➔ ♥❣æ♥ ♥❣ú tr➯♥ ♠å✐ ❜↔♥❣ ❝❤ú✳ ❚❛ ♣❤➙♥ ❜✐➺t ♥❣æ♥ ♥❣ú ré♥❣ L = ∅ ❦❤→❝ ợ ổ ỳ ỗ ởt tỷ rộ L = {ε}✳ ❱➼ ❞ư ✶✳✹✳ ✰ Σ∗ ❧➔ ♥❣ỉ♥ ♥❣ú ỗ tt tứ tr + ổ ỳ ỗ tt tứ rộ tr Σ✳ ✰ A = {a, b, c}✱ B = {aa, bb, cc, ab, bc, ca}✱ C = {abca, cba, a, bc}✱ D = {abccda, acbab, abcc, ac} ❧➔ ♥❣æ♥ ♥❣ú tr➯♥ ❜↔♥❣ ❝❤ú Σ = {a, b, c}✳ ✰ L1 = {0}✱ L2 = {000, 10, 1111, 110}✱ L3 = {0100, 01010, 100, 01010} ❧➔ ❝→❝ ♥❣æ♥ ♥❣ú tr➯♥ ❜↔♥❣ ❝❤ú Γ = {0, 1}✳ ❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ ♥❣æ♥ ♥❣ú✿ ❉♦ ♠é✐ ♥❣ỉ♥ ♥❣ú ❧➔ ♠ët t➟♣ ❤đ♣ ♥➯♥ t❛ ❝â ❝→❝ ♣❤➨♣ t♦→♥ ✤↕✐ sè t➟♣ ❤đ♣ ♥❤÷ ❧➔ ♣❤➨♣ ❣✐❛♦✱ ♣❤➨♣ ✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ Pữỡ ỵ ợ ổ ỳ ✤÷đ❝ s✐♥❤ ❜ð✐ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤ì♥ ✤à♥❤ ❧➔ trò♥❣ ợ ợ ổ ỳ ữủ s tt ỳ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t✿ LN ❧➔ ❧ỵ♣ ♥❣ỉ♥ ♥❣ú s✐♥❤ ❜ð✐ ❝→❝ ♦t♦♠❛t ❤ú✉ ❤↕♥ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤❀ LD ❧➔ ❧ỵ♣ ♥❣ỉ♥ ♥❣ú s✐♥❤ ❜ð✐ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤ì♥ ✤à♥❤✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ LN = LD ❜➡♥❣ ❝→❝❤ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ❤❛✐ ❜❛♦ ❤➔♠ t❤ù❝✿ • LN ⊆ LD ✳ ●✐↔ sû L ❧➔ ♠ët ♥❣æ♥ ♥❣ú tũ ỵ tở ợ LN tỗ t ♠ët ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ A ✤♦→♥ ♥❤➟♥ L✱ tù❝ t õ T (A) = L ỵ ự tr tỗ t ởt tt ỡ ✤à♥❤ M s❛♦ ❝❤♦ L = T (M )✳ ❱➟② L t❤✉ë❝ ❧ỵ♣ LD ✱ ❤❛② LN ⊆ LD ✳ • LD ⊆ LN ✳ ●✐↔ sû L ❧➔ ♠ët ợ ổ ỳ tũ ỵ tở ợ LD tỗ ởt tt ỡ M L✱ ♥❣❤➽❛ ❧➔ t❛ ❝â T (M ) = L✳ ❚✉② ♥❤✐➯♥ t❛ ❝â t❤➸ ❝♦✐ ❤➔♠ ❝❤✉②➸♥ ❝õ❛ ♦t♦♠❛t ✤ì♥ ✤à♥❤ ❧➔ tr÷í♥❣ ❤đ♣ ✤ì♥ ❣✐↔♥ ❝õ❛ ❤➔♠ ❝❤✉②➸♥ tr♦♥❣ ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ❤❛② ♥â✐ ❝→❝❤ ❦❤→❝ t❤➻ ♦t♦♠❛t ✤ì♥ ✤à♥❤ ❧➔ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤✳ ❱➻ t❤➳ t❛ ❝â t❤➸ ❝♦✐ L ✤÷đ❝ ✤♦→♥ ♥❤➟♥ ❜ð✐ ♦t♦♠❛t ❤ú✉ ❤↕♥ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ✤➦❝ ❜✐➺t✱ tù❝ ❧➔ LD ⊆ LN ✳ ✷✳✺ ▼ët ù♥❣ ❞ư♥❣ ❝õ❛ ♦t♦♠❛t ❤ú✉ ❤↕♥ ▼ët ✈➜♥ ✤➲ t❤÷í♥❣ ❣➦♣ ❝õ❛ ♥❣÷í✐ ❞ò♥❣ ♠→② t➼♥❤ ✈➔ ♠↕♥❣ ■♥t❡r♥❡t ❧➔ t ỳ ỗ ữ trỳ ợ ổ t ữ t õ t❤➸ t➻♠ t❤➜② ♠ët tø ❤♦➦❝ ❝ö♠ tø tr♦♥❣ ♠ët ✈➠♥ ❜↔♥✳ ✸✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ❍❛② ❧➔ ❝ỉ♥❣ ❝ư t➻♠ ❦✐➳♠ ●♦♦❣❧❡✱ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ tr❛♥❣ ✇❡❜ ❧✐➯♥ q✉❛♥ ✤➳♥ t❤ỉ♥❣ t✐♥ ♠➔ t❛ ❝➛♥ ❜➡♥❣ ❝→❝❤ ♥❤➟♣ ✈➔♦ ❝→❝ tø ❦❤â❛✳ ◆❤ú♥❣ ❝ỉ♥❣ ❝ư t➻♠ ❦✐➳♠ ♥❤÷ t❤➳ ❧➔ ♠ët ✈➼ ❞ö ✤✐➸♥ ❤➻♥❤ ✈➲ sû ❞ö♥❣ ♦t♦♠❛t ❤ú✉ ❤↕♥ tr số sỷ t õ t ỵ tü ✭❤❛② ♠ët tø✮✱ ❣å✐ t➢t ❧➔ tø ❦❤â❛✱ t❛ t ú tr ỳ trữợ ✤â✱ ♥➳✉ t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ♠ët ♦t♦♠❛t ❤ú✉ ❤↕♥ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ✤♦→♥ ♥❤➟♥ ✤÷đ❝ tø ❦❤â❛ ✤â✳ ❚❛ ♥❤➟♣ ❝❤✉é✐ ❦➼ tü ❝➛♥ t➻♠✱ ❝â ♠ët ♦t♦♠❛t ❤ú✉ ❤↕♥ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ✤ì♥ ❣✐↔♥ ✤➸ ✤♦→♥ ♥❤➟♥ ❝❤✉é✐ ỵ tỹ õ ữủ ỹ ữ s õ ♠ët tr↕♥❣ t❤→✐ ❜➢t ✤➛✉ q0 ✈➔ ♠ët ❤➔♠ ❝❤✉②➸♥ tr ỳ t ú ỵ tr↕♥❣ t❤→✐ ❜➢t ✤➛✉ t❤➸ ❤✐➺✉ ❜➢t ✤➛✉ ✧sü ✤♦→♥ ♥❤➟♥✧✱ ♥➳✉ ❝❤÷❛ t❤➜② ♠ët tr♦♥❣ ❝→❝ tø ❦❤â❛ t❤➻ ❝❤÷❛ ❜➢t ✤➛✉ ✭❦➸ ❝↔ ❧➔ ❦❤✐ t❤❛② ✤ê✐ ♠ët sè ❝❤ú ❝→✐ ❝õ❛ tø ❤♦➦❝ ❧➔ t❤ù tü ❝õ❛ ú ợ tứ õ a1 a2 ✳✳✳✱ ak s➩ ❝â k tr↕♥❣ t❤→✐ ❧➔ q1 ✱ q2 ✱ ✳✳✳✱ qk t÷ì♥❣ ù♥❣✳ ❚ø tr↕♥❣ t❤→✐ ❜➢t q0 s tr t q1 ỵ tü a1 ✱ tr↕♥❣ t❤→✐ q1 ❝❤✉②➸♥ s❛♥❣ tr↕♥❣ t❤→✐ q2 ỵ tỹ a2 ự t tử q✉→ tr➻♥❤ ♥❤÷ ✈➙②✱✳✳✳ ❑❤✐ ❣➦♣ tr↕♥❣ t❤→✐ qk ✲ tr↕♥❣ t❤→✐ ✤÷đ❝ ✤♦→♥ ♥❤➟♥ ✭tr↕♥❣ t❤→✐ ❦➳t t❤ó❝✮ ✈➔ ❦➳t ❧✉➟♥ tø ❦❤â❛ a1 a2 ak ✤÷đ❝ t➻♠ t❤➜②✳ ❚✉② ✈➟②✱ tr➯♥ t❤ü❝ t➳ ❝→❝ ♣❤➛♥ ♠➲♠ ①û ❧➼ ✈➠♥ ❜↔♥ t❤÷í♥❣ ♣❤↔✐ ❝❤ù❛ ♥❤✐➲✉ ❞↕♥❣ ❧➺♥❤ ♣❤ù❝ t↕♣ ỡ ỗ ỳ ợ tr s sõt ❦➳t q✉↔ t➻♠ ✤÷đ❝✳ ❈❤ó♥❣ t❛ ✤÷❛ r❛ ❤❛✐ ❝→❝❤ ①û ❧➼ s❛✉ ✤➸ ❤♦➔♥ t❤✐➺♥ ❤ì♥ ❝ỉ♥❣ ❝ư ♥➔②✿ ✶✳ ❱✐➳t ♠ët ❝❤÷ì♥❣ tr➻♥❤ ♠ỉ ♣❤ä♥❣ ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ◆❋❆ ✸✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ♥➔② ❜➡♥❣ ❝→❝❤ ①→❝ ✤à♥❤ ❝→❝ t➟♣ tr↕♥❣ t❤→✐ ❝õ❛ ♥â s❛✉ ❦❤✐ ✤å❝ ❦➼ ❤✐➺✉ ✤➛✉ ✈➔♦✳ ỗ t s ỗ t ❝❤✉②➸♥ tr➯♥ ①→❝ ✤à♥❤ ♠ët ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ◆❋❆ ✤♦→♥ ♥❤➟♥ ❝→❝ tø ❦➳t t❤ó❝ ❜➡♥❣ ✵✶✳ ❘ã r➔♥❣ ❤ì♥✱ t❛ ❝â t❤➸ ♠ỉ t↔ ✤÷đ❝ ♦t♦♠❛t ✤♦→♥ ♥❤➟♥ tứ 00101 q sỡ ỗ tt ỡ tữỡ ữỡ ợ tt ổ ỡ ✤➣ ❝❤♦ t❤❡♦ ♥❣✉②➯♥ t➢❝ ✰ ❚r↕♥❣ t❤→✐ ❦❤ð✐ ✤➛✉ q0 ❝õ❛ ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ◆❋❆ ❝ơ♥❣ ❧➔ tr↕♥❣ t❤→✐ ❦❤ð✐ ✤➛✉ ❝õ❛ ♦t♦♠❛t ✤ì♥ ✤à♥❤ ❉❋❆ t÷ì♥❣ ✤÷ì♥❣❀ ✰ ❚r↕♥❣ t❤→✐ p ❦❤→❝ ❝õ❛ ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ◆❋❆ ❧➔ tr↕♥❣ t❤→✐ ✤✐ tø tr↕♥❣ t❤→✐ ❦❤ð✐ ✤➔✉ q0 ✤✐ q✉❛ ❝→❝ ❦➼ ❤✐➺✉ a1 a2 am ✳ ❑❤✐ ✤â ♠ët tr↕♥❣ t❤→✐ ❝õ❛ ♦t♦♠❛t ✤ì♥ ✤à♥❤ ❉❋❆ t÷ì♥❣ ✤÷ì♥❣ ❧➔ t➟♣ ❝→❝ tr↕♥❣ t❤→✐ ❝õ❛ ♦t♦♠❛t ❦❤ỉ♥❣ ỡ ỗ q0 p tr↕♥❣ t❤→✐ ✸✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ❝õ❛ ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ◆❋❆ ✤✐ tø q0 t❤❡♦ ❝♦♥ ✤÷í♥❣ ❝â ♥❤➣♥ ❧➔ ❤➟✉ tè ❝õ❛ a1 a2 am ✳ ❙❛✉ ✤â ♠æ ❤➻♥❤ ❤â❛ trü❝ t✐➳♣ ♦t♦♠❛t ✤ì♥ ✤à♥❤ ❉❋❆ ✤â✳ ❱➼ ❞ư ✷✳✽✳ ●✐↔ sû ♠✉è♥ ✤♦→♥ ♥❤➟♥ tø abcd ✈➔ tø mnp t❤➻ t❛ ❝â t❤➸ ①➙② ❞ü♥❣ ♠ët ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ỹ ữỡ tr t q ỗ t❤à ❝❤✉②➸♥ s❛✉ tr♦♥❣ ✤â✿ ✰ Σ ❧➔ t➟♣ t➜t ỵ tỹ q0 ✲ tr↕♥❣ t❤→✐ ❜❛♥ ✤➛✉✳ ✰ q1 ✤➳♥ q4 ✲ tr↕♥❣ t❤→✐ ✤➸ ✤♦→♥ ♥❤➟♥ tø abcd✳ ✰ q5 ✤➳♥ q7 ✲ tr↕♥❣ t❤→✐ ✤➸ ✤♦→♥ ♥❤➟♥ tø mnp✳ ❚ø ✤â✱ t❛ ①➙② ❞ü♥❣ ♠ët ♦t♦♠❛t ✤ì♥ ✤à♥❤ ❉❋❆ ♥❤÷ ỗ t s õ tốt ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ❚r♦♥❣ ❤❛✐ ❝→❝❤ ♥➯✉ tr➯♥✱ ❝→❝❤ ✤÷❛ ✈➲ ♦t♦♠❛t ✤ì♥ ✤à♥❤ ❉❋❆ t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ ♥❤✐➲✉ ❤ì♥ ❞♦ ♥❣÷í✐ t❛ t❤➜② r➡♥❣ sè ❧÷đ♥❣ ❝→❝ tr↕♥❣ t❤→✐ ❝õ❛ ♦t♦♠❛t ✤ì♥ ✤à♥❤ ❉❋❆ t÷ì♥❣ ✤÷ì♥❣ t❤➻ ❦❤ỉ♥❣ ❜❛♦ ❣✐í ❧ỵ♥ ❤ì♥ sè tr↕♥❣ t❤→✐ ❝õ❛ ♦t♦♠❛t ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ◆❋❆✳ ✸✺ ❈❤÷ì♥❣ ✸ ◆●➷◆ ◆●Ú ❈❍➑◆❍ tt ỳ ữủ ợ t ữỡ ữ r ự ổ ỳ ợ ổ ♥❣ú ✤â ❝❤➼♥❤ ❧➔ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉② ✤÷đ❝ t➻♠ ❤✐➸✉ s❛✉ ✤➙②✳ ✸✳✶ ❇✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳ ❈❤♦ ❜↔♥❣ ❝❤ú Σ = {a1 , a2 , , an }✱ ❦❤✐ ✤â ❜✐➸✉ ❝❤➼♥❤ q✉② ✭r❡❣✉❧❛r ❡①♣r❡s✐♦♥s✮ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ✤➺ q✉② ♥❤÷ s❛✉✿ t❤ù❝ ✶✳ ∅ ✈➔ a ✭✈ỵ✐ a ∈ Σ✮ ❧➔ ❝→❝ ❜✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② tr➯♥ ❜↔♥❣ ❝❤ú Σ ❜✐➸✉ ❞✐➵♥ ♥❣æ♥ ♥❣ú ∅ ✈➔ ♥❣æ♥ ♥❣ú {a}✳ ✷✳ ◆➳✉ r ✈➔ s ❧➔ ❤❛✐ ❜✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② ❜✐➸✉ ❞✐➵♥ ❝→❝ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② R ✈➔ S tr➯♥ ❜↔♥❣ ❝❤ú Σ t❤➻✿ • r + s ❧➔ ❜✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② tr➯♥ ❜↔♥❣ ❝❤ú Σ ❜✐➸✉ ❞✐➵♥ ♥❣æ♥ ♥❣ú R ∪ Q❀ • r.s ❧➔ ❜✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② tr➯♥ ❜↔♥❣ ❝❤ú Σ ❜✐➸✉ ❞✐➵♥ ♥❣ỉ♥ ♥❣ú R.S ❀ • r+ ✭❤❛② s+ ✮ ❧➔ ❜✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② tr➯♥ ❜↔♥❣ ❝❤ú Σ ❜✐➸✉ ❞✐➵♥ ♥❣æ♥ ♥❣ú R+ ✭❤❛② S + ✮✳ ✸✻ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ✸✳ ❑❤ỉ♥❣ ❝â ❝→❝ ❜✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② ♥➔♦ ❦❤→❝ tr➯♥ ❜↔♥❣ ❝❤ú Σ ♥❣♦➔✐ ❝→❝ ❜✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr➯♥✳ ❇✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr➯♥ ❞ò♥❣ ✤➸ ❜✐➸✉ ✤↕t ❝→❝ ❧♦↕✐ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉② ✤÷đ❝ ♠ỉ t↔ s❛✉ ✤➙②✳ ✸✳✷ ◆❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳ ❈❤♦ ❜↔♥❣ ❝❤ú Σ = {a1 , a2 , , an }✱ ❦❤✐ ✤â ♥❣ỉ♥ ❝❤➼♥❤ q✉② ✭r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s✮ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ✤➺ q✉② ♥❤÷ s❛✉✿ ♥❣ú ✶✳ ∅ ✈➔ {ai } ✭i = 1, 2, , n✮ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② tr➯♥ ❜↔♥❣ ❝❤ú Σ✳ ✷✳ ◆➳✉ R ✈➔ S ❧➔ ❤❛✐ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② tr➯♥ ❜↔♥❣ ❝❤ú Σ t❤➻ R ∪ S ❀ R.S ❀ R+ ✭❤❛② S + ✮ ❧➔ ❝→❝ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② tr➯♥ ❜↔♥❣ ❝❤ú Σ✳ ✸✳ ❑❤æ♥❣ ❝â ❝→❝ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② ♥➔♦ ❦❤→❝ tr➯♥ ❜↔♥❣ ❝❤ú Σ ♥❣♦➔✐ ❝→❝ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr➯♥✳ ◆❤➟♥ ①➨t✿ ❚r♦♥❣ ❈❤÷ì♥❣ ✶ t❛ ✤➣ ✤à♥❤ ♥❣❤➽❛ ✈➲ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② q✉❛ ✈➠♥ ♣❤↕♠ ❝❤➼♥❤ q✉②✱ ❝â t❤➸ t❤➜② r➡♥❣ ✤à♥❤ ♥❣❤➽❛ ♥➔② ✈ỵ✐ ✣à♥❤ ♥❣❤➽❛ ✸✳✷ tr➯♥ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣ ♥❤❛✉✳ ❳➨t ❤❛✐ ✈➠♥ ♣❤↕♠ G1 = Σ, {S}, S, {S → aS | a ∈ Σ} ✱ G2 = Σ, {S}, S, {S → | ∈ Σ} ✱ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② s✐♥❤ ❜ð✐ ✈➠♥ ♣❤↕♠ ♥➔② ❧➛♥ ❧÷đt ❧➔ ∅ = L(G1 )✱ {ai } = L(G2 )✳ ▼➦t ❦❤→❝✱ t❛ õ t ự ữủ ợ ổ ỳ q✉② ✤â♥❣ ✈ỵ✐ ♣❤➨♣ t♦→♥ ❤đ♣✱ ♥❤➙♥ ❣❤➨♣ ✈➔ ❧➦♣✳ ữ ợ ổ ỳ q tr ð tr➯♥ trò♥❣ ✈ỵ✐ ✤à♥❤ ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✷✳ ✣à♥❤ ỵ ổ ỳ q tr ỳ Σ ✤➲✉ ♥❤➟♥ ✸✼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ✤÷đ❝ tø ❝→❝ ♥❣ỉ♥ ♥❣ú ❤ú✉ ❤↕♥ ❜➡♥❣ ❝→❝❤ →♣ ❞ö♥❣ ♠ët sè ❤ú✉ ❤↕♥ ❧➛♥ ❝→❝ t ủ ú ỵ ❈→❝ ✈➠♥ ♣❤↕♠ ❝❤➼♥❤ q✉② ❦❤æ♥❣ ❝❤ù❛ ❝→❝ q✉② t➢❝ s✐♥❤ tø ré♥❣ ♥➯♥ ❝→❝ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉② ❝ơ♥❣ ❦❤æ♥❣ ❝❤ù❛ tø ré♥❣✳ ◆❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② s✉② rë♥❣ ❧➔ ❝→❝ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② ❝❤ù❛ tø ré♥❣ ε✱ ✈➠♥ ♣❤↕♠ ❝❤➼♥❤ q✉② ❝❤ù❛ q✉② t➢❝ ré♥❣ ❧➔ ✈➠♥ q s rở ỵ ổ ♥❣ú tr➯♥ ❜↔♥❣ ❝❤ú Σ ❧➔ ❝❤➼♥❤ q✉② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜➡♥❣ ♠ët ❜✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉②✳ ❱➼ ❞ö ✸✳✶✳ ❈❤♦ ❜✐➸✉ t❤ù❝ r = (01∗ + 02)1✳ ❳→❝ ✤à♥❤ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉② ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ ❜✐➸✉ t❤ù❝ tr➯♥✳ ❚❛ ❝â✿ r = (01∗ + 02)1 = 01∗ + 021 ❱➟② ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② ❜✐➸✉ ❞✐➵♥ ❜ð✐ r ❧➔✿ L (r) = L (01∗ + 021) = L (01∗ 1) ∪ L (021) = {01n , 021 | n ≥ 1} ✸✳✸ ❙ü ❧✐➯♥ ❤➺ ❣✐ú❛ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✈➔ ổ ỳ q ỵ ởt ổ ỳ q t tỗ t ởt tt ❤ú✉ ❤↕♥ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ A ✤ì→♥ ♥❤➟♥ L✱ tù❝ ❧➔ L = T (A)✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû L ❧➔ ♠ët ♥❣æ♥ ♥❣ú s✐♥❤ ❜ð✐ ✈➠♥ ♣❤↕♠ ❝❤➼♥❤ q✉② G = Σ, ∆, S, P ✱ ❤❛② L = L(G)✳ ❳➨t ♦t♦♠❛t ❤ú✉ ❤↕♥ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ A = Q, Σ, δ, q0 , F ✱ tr♦♥❣ ✤â✿ ✸✽ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ✰ Q = {E} ợ E ỵ ♠ỵ✐ ✈➔ E ∈ / Σ ∪ ∆❀ ✰ q0 = S ❀ ✰ F = {E} ♥➳✉ q✉② t➢❝ S → ε ∈ / P ✈➔ F = {E, S} ♥➳✉ S → ε ∈ P ❀ ✰ ∀A ∈ ∆✱ ∀a ∈ Σ t❛ ✤➦t δ(A, a) = {B ∈ ∆ | A → aB ∈ P } ∪ {E | A → a ∈ P } ✈➔ δ(E, a) = ∅✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ L = T (A) t ý L rữớ ủ = õ tỗ t q t➢❝ S → ε ∈ P ❤❛② ♦t♦♠❛t A ❝â t➟♣ ❦➳t t❤ó❝ F = {E, S}✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ t➟♣ ❦➳t t❤ó❝ t❛ ❝â δ(S, ε) = {S} ❞♦ ✤â δ(S, ε)∩F = ∅✳ ❙✉② r❛ ε ∈ T (A) T (A) rữớ ủ ω = a1 a2 an = ε✳ ❑❤✐ ✤â t❛ ❝â ❞➝♥ s✉➜t S a1 a2 an−1 An−1 A1 a2 A2 ✱ ✳✳✳✱ Ai−1 ω ❤❛② s✉② ❞➝♥ S a1 A a1 a2 A a1 a2 an ✳ ❚ø ✤â✱ t❛ ❝â ❝→❝ q✉② t➢❝ S Ai ✱ ✳✳✳✱ An−1 a1 A1 ✱ an t❤✉ë❝ P ✳ ❚ø ❝→❝❤ ①→❝ ✤à♥❤ ♦t♦♠❛t A✱ s✉② r❛ A1 ∈ δ(S, a1 )✱ A2 ∈ δ(A1 , a2 )✱✳✳✱E ∈ δ(An−1 , an )✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ t➟♣ ❦➳t t❤ó❝✱ E ∈ δ(S, ω)✳ ❙✉② r❛ ω ∈ T (A)✳ ❱➻ ✈➟② L ⊂ T (A) ✭✶✮ ✷✳ ▲➜② t ý T (A) rữớ ủ ω = ε✳ ❑❤✐ ✤â δ(S, ε) ∩ F = ∅✳ ▼➔ t❤❡♦ ❝→❝❤ ①→❝ ✤à♥❤ ♦t♦♠❛t A t❤➻ δ(S, ) = {S} S F tỗ t q✉② t➢❝ S → ε ∈ P ✳ ❙✉② r❛ L(G) rữớ ủ = a1 a2 an = ε✳ ✸✾ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ❑❤✐ ✤â E ∈ δ(S, ω)✱ õ tỗ t tr t A1 , A2 , , An−1 ∈ ∆ s❛♦ ❝❤♦ A1 ∈ δ(S, a1 )✱ A2 ∈ δ(A1 , a2 )✱✳✳✱E ∈ δ(An−1 , an )✳ ❚ø ✤â✱ t❛ ❝â ❞➣② ❝→❝ s✉② ❞➝♥ S ✳✳✳✱ An−1 a1 A1 ✱ A1 an t❤✉ë❝ P ❤❛② ❞➣② s✉② ❞➝♥ S a1 a2 an−1 An−1 a2 A2 ✱ ✳✳✳✱ Ai−1 Ai ✱ a1 A1 a1 a2 A2 a1 a2 an = ω ✳ ❙✉② r❛ ω ∈ L ❤❛② T (A) ⊆ L ✭✷✮ ❚ø ✭✶✮ ✈➔ ✭✷✮ s✉② r❛ L = T (A) ✭✤♣❝♠✮✳ ❱➼ ❞ư ✸✳✷✳ ❈❤♦ ♥❣ỉ♥ ♥❣ú L = {an bm | n, m ≥ 1}✳ ❚❛ ❝â ✈➠♥ ♣❤↕♠ ❝❤➼♥❤ q✉② G = {a, b} , {S, A, B} , S, P tr♦♥❣ ✤â S → aA✱ S → aS ✱ A → aB ✱ B → bB ✱ B → B ✱ A → A✳ ❚❤❡♦ ✤à♥❤ ỵ tr t ỹ ổt tt ỳ ổ ✤ì♥ ✤à♥❤ A = {S, A, B, E} , {a, b}, δ, S, E ✱ tr♦♥❣ ✤â δ(S, a) = {A, S}❀ δ(A, a) = {A, B}❀ δ(B, b) = {B, E} ỗ t A ữ s ứ ✤â t❛ t❤➜② T (A) = {an bm | n, m ≥ 1} = L✱ tù❝ ❧➔ ♦t♦♠❛t A ✤♦→♥ ổ ỳ q L ỵ L ❧➔ ♠ët ♥❣ỉ♥ ♥❣ú ✤÷đ❝ ✤♦→♥ ♥❤➟♥ ❜ð✐ ♠ët ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤ì♥ ✤à♥❤ t❤➻ L ❧➔ ♠ët ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉②✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤ì♥ ✤à♥❤ M = Q, Σ, δ, q0 , F ✳ L ❧➔ ♥❣ỉ♥ ♥❣ú ✤÷đ❝ ✤♦→♥ ♥❤➟♥ ❜ð✐ M ✱ tù❝ ❧➔ L = T (M )✳ ❳➨t ✈➠♥ ♣❤↕♠ ❝❤➼♥❤ q✉② G = Σ, Q, q0 , P tr♦♥❣ ✤â ❝→❝ q✉② t➢❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ P = {q → ap | δ(q, a) = p} ∪ {q → a | δ(q, a) ∈ F }✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ L(G) = L ✹✵ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ✰ ❚r÷í♥❣ ❤đ♣ ✶✿ ε ∈ /L • ▲➜② ω = a1 a2 an ∈ L(G) t ý õ G tỗ t s✉② ❞➝♥ q0 q0 a1 q a1 a2 q ω ❤❛② ❞➣② ❝→❝ s✉② ❞➝♥ a1 a2 an−1 qn−1 a1 a2 an = ω ❚r♦♥❣ ✤â P tỗ t q t q0 a1 q1 q1 → a2 q2 ✱ ✳✳✳✱ qn−2 → an−1 qn−1 ✱ qn−1 → an ✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ δ(q0 , a1 ) = q1 ✱ δ(q1 , a2 ) = q2 ✱ ✳✳✳✱ δ(qn−2 , an−1 ) = qn−1 ✱ δ(qn−1 , an ) ∈ F ✳ ❚ù❝ ❧➔ δ(q0 , ω) ∈ F ❤❛② ω ∈ T (M ) = L✳ ❙✉② r❛ L(G) ⊆ T (M ) ✭✶✮ • ▲➜② ω = a1 a2 an ∈ T (M ) t ý õ tỗ t tr t q0 ✱ q1 ✱ ✳✳✳✱ qn s❛♦ ❝❤♦ δ(q0 , a1 ) = q1 ✱ δ(q1 , a2 ) = q2 ✱ ✳✳✳✱ δ(qn−2 , an−1 ) = qn−1 ✱ δ(qn−1 , an ) ∈ F ✳ ❉♦ ✤â tr♦♥❣ P ❝â ❝→❝ q✉② t➢❝ q0 → a1 q1 ✱ q1 → a2 q2 ✱ ✳✳✳✱ qn−2 → an−1 qn−1 ✱ qn−1 → an ✳ ❙✉② r❛ t❛ ❝â ❞➣② ❝→❝ s✉② ❞➝♥ q0 a1 q ❤❛② ✈✐➳t ❣å♥ ❧➔ q0 a1 a2 q a1 a2 an−1 qn−1 a1 a2 an = ω ω ✳ ❙✉② r❛ ω ∈ L(G)✳ ❙✉② r❛ T (M ) ⊆ L(G) ✭✷✮ ❚ø ✭✶✮ ✈➔ ✭✷✮ t❛ ❝â T (M ) = L(G) ✰ ❚r÷í♥❣ ❤đ♣ ✷✿ ε ∈ L ❉ü❛ ✈➔♦ ✈➠♥ ♣❤↕♠ G tr➯♥✱ t❛ ①➙② ❞ü♥❣ ✈➠♥ ♣❤↕♠ G = Σ, Q, q0 , P ✈ỵ✐ P = {q0 → ε}∪{q → ap | δ(q, a) = p = q0 }∪{q → a | δ(q, a) ∈ F }✳ ✹✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ❑❤✐ ✤â t❛ ❝â L(G ) = L(G) ∪ {} t ủ ợ trữớ ủ t L(G ) = T (M ) ỵ ữủ ự ❱➼ ❞ư ✸✳✸✳ ❈❤♦ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤ì♥ ✤à♥❤ M = {q0 , q1 }, {a, b}, δ, q0 , {q0 } tr♦♥❣ ✤â δ(q0 , a) = q1 ✱ δ(q1 , a) = q0 ✱ δ(q1 , b) = q1 ỗ t M t r T (M ) = {ω1 ω2 ωn | ωi = abi a, ≤ i ≤ n}✳ T (M ) ❧➔ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② s✐♥❤ ❜ð✐ ✈➠♥ ♣❤↕♠ G = {a, b}, {q0 , q1 }, q0 , P ✈ỵ✐ P ❝â ❝→❝ q✉② t➢❝ q0 → aq1 ✱ q1 → aq0 ✱ q1 → bq1 ✳ ◆❤➟♥ ①➨t✿ ỵ ự t ữ r ❧✐➯♥ ❤➺ ❣✐ú❛ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✈➔ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q ữ s D ợ ổ ♥❣ú ✤♦→♥ ♥❤➟♥ ❜ð✐ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤ì♥ ✤à♥❤ ✭❉❋❆✮❀ • N ❧➔ ❧ỵ♣ ❝→❝ ♥❣ỉ♥ ♥❣ú ✤♦→♥ ♥❤➟♥ ❜ð✐ ♦t♦♠❛t ❤ú✉ ❤↕♥ ❦❤ỉ♥❣ ✤ì♥ ✤à♥❤ ✭◆❋❆✮❀ • R ❧➔ ❧ỵ♣ ❝→❝ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉②✳ ❚❛ ❝â✿ ✰ ❚❤❡♦ ỵ D = N ỵ R N ỵ ✸✳✹✱ D ⊂ R✳ ✹✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ❙✉② r❛ D = N = R✳ ❱➟② ❧ỵ♣ ♥❣ỉ♥❣ ♥❣ú ♠➔ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤♦→♥ ổ ỳ q ú ỵ õ ❦❤→❝ t❤➻ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ L ❧➔ ♠ët ♥❣æ♥ ♥❣ú q r ỗ t↕✐ ♠ët ❜✐➸✉ t❤ù❝ ❝❤➼♥❤ q✉② ❜✐➸✉ ❞✐➵♥ L✳ ✷✳ ỗ t ởt q s r ổ ỳ L ỗ t ởt tt ỳ ♥❤➟♥ L✳ ✹✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲➯ ▼✐♥❤ P❤÷ì♥❣ ❑➌❚ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ ♥➔② ✤➣ ✤✐ t➻♠ ❤✐➸✉ ✈➲ ✧❖t♦♠❛t ❤ú✉ ❤↕♥ ✈➔ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉②✧ tr♦♥❣ ❜❛ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ❝✉♥❣ ❝➜♣ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ♥❤÷ ❜↔♥❣ ❝❤ú✱ tø✱ ♥❣ỉ♥ ♥❣ú ✈➔ ✈➠♥ ♣❤↕♠ ❧➔♠ t✐➲♥ ✤➲ ❝❤♦ ❝→❝ ✈➜♥ ✤➲ ð ❝❤÷ì♥❣ s❛✉✳ ❈❤÷ì♥❣ ✷ t➻♠ ❤✐➸✉ ✈➲ ❦❤→✐ ♥✐➺♠✱ ♣❤➙♥ ❧♦↕✐✱ ❤➻♥❤ t❤ù❝ ❤♦↕t ✤ë♥❣ ✈➔ ♠æ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ ♦t♦♠❛t ❤ú✉ ❤↕♥ ❝ò♥❣ ✈ỵ✐ ✤â ❧➔ ❧ỵ♣ ♥❣ỉ♥ ỳ tt ỳ ỵ ỵ ự r ợ ổ ♥❣ú ♠➔ ❤❛✐ ❧♦↕✐ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤♦→♥ ♥❤➟♥ ❧➔ trò♥❣ ♥❤❛✉✳ ❚r➯♥ ❝ì sð ✤â✱ ❝❤÷ì♥❣ ✸ ✤➣ tr➻♥❤ ❜➔② ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② t❤æ♥❣ ❦❤→✐ ♥✐➺♠ ✤➺ q✉② tữỡ ữỡ ợ q ữỡ ỵ ❝❤♦ t❤➜② ♠è✐ q✉❛♥ ❤➺ ♠➟t t❤✐➳t ❣✐ú❛ ♥❣æ♥ ♥❣ú q tt ỳ ợ ổ ỳ ữủ ♦t♦♠❛t ❤ú✉ ❤↕♥ ✤♦→♥ ♥❤➟♥ ❝❤➼♥❤ ❧➔ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉②✱ ❝❤ù❝ ♥➠♥❣ ❝❤➼♥❤ ❝õ❛ ♦t♦♠❛t ❤ú✉ ❤↕♥✳ ❉♦ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❤➭♣ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✈➝♥ ❝á♥ ♥❤✐➲✉ t❤✐➳✉ sât ♠♦♥❣ t❤➛② ❝æ õ ỵ t ỡ t❤❛♠ ❦❤↔♦ ❬✶❪ ✣♦➔♥ ❱➠♥ ❇❛♥✱ ◆❣✉②➵♥ ❍✐➲♥ ❚r✐♥❤ ✭✷✵✶✵✮✱ ●✐→♦ tr➻♥❤ ♥❣æ♥ ♥❣ú ❤➻♥❤ t❤ù❝ ✈➔ ætæ♠→t✱ ◆❳❇ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ✣à♥❤ ✭✷✵✶✷✮✱ ●✐→♦ tr➻♥❤ ➷tæ♠→t ✈➔ ◆❣æ♥ ♥❣ú ❤➻♥❤ t❤ù❝✱ ◆❳❇ ✣↕✐ ❤å❝ ◆æ♥❣ ♥❣❤✐➺♣✳ ❬✸❪ ❏✳ ❊✳ ❍♦❝r♦❢t✱ ❘✳ ▼♦t✇❛♥✐✱ ❏✳ ❉✳ ❯❧❧♠❛♥ ✭✷✵✵✶✮✱ ■♥tr♦❞✉t✐♦♥ t♦ ❆✉t♦♠❛t❛ ❚❤❡♦r② ▲❛♥❣✉❛❣❡s✱ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ❆❞❞✐s✐♦♥ ✲ ❲❡s✲ ❧❡② P✉❜❧✐s❤✐♥❣ ❈♦♠♣❛♥②✳ ■♥❝✳ ❬✹❪ ✣➦♥❣ ❍✉② ❘✉➟♥ ✭✷✵✵✷✮✱ ỵ tt ổ ỳ tự tt ❍➔ ◆ë✐✳ ❬✺❪ P❤❛♥ ❚❤à ❚÷ì✐ ✭✶✾✽✻✮✱ ❚r➻♥❤ ❜✐➯♥ ❞à❝❤✱ ◆❳❇ ●✐→♦ ❞ö❝✳ ✹✺ ...TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* LÊ MINH PHƯƠNG OTOMAT HỮU HẠN VÀ NGÔN NGỮ CHÍNH QUY KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chun ngành: Toán ứng dụng Người hướng dẫn khoa