TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* BÙI PHƯƠNG HIỀN VĂN PHẠM VÀ NGÔN NGỮ PHI NGỮ CẢNH KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Toán ứng dụng HÀ NỘI – 2018 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* BÙI PHƯƠNG HIỀN VĂN PHẠM VÀ NGƠN NGỮ PHI NGỮ CẢNH 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✐♥ ❜➔② tä ỏ ỡ tợ t ổ rữớ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ tê ❜ë ♠ỉ♥ ❚♦→♥ ù♥❣ ❞ư♥❣ ❝ơ♥❣ ♥❤÷ ❝→❝ t❤➛② ❝æ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ t ỳ tr tự qỵ t t❤✉➟♥ ❧đ✐ ✤➸ ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ♥❤✐➺♠ ✈ư ❦❤â❛ ❤å❝ ✈➔ ❦❤â❛ ❧✉➟♥✳ ✣➦❝ ❜✐➺t✱ ❡♠ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ữ ữớ trỹ t ữợ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ✤➸ ❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❉♦ t❤í✐ ❣✐❛♥✱ ♥➠♥❣ ❧ü❝ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❜↔♥ ❦❤â❛ ❧✉➟♥ ❝á♥ ♥❤✐➲✉ s❛✐ sât✳ ❱➻ ✈➟②✱ ❡♠ r➜t ữủ ỳ ỵ õ ỵ qỵ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ❇ò✐ P❤÷ì♥❣ ❍✐➲♥ ✶ ▲❮■ ữợ sỹ ữợ ❝õ❛ ❚❙✳ ❑✐➲✉ ❱➠♥ ❍÷♥❣ ❦❤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✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ❇ò✐ P❤÷ì♥❣ ❍✐➲♥ ✷ ▼ư❝ ❧ư❝ ▲í✐ ♥â✐ ✤➛✉ ✺ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✻ ✶✳✶ ✶✳✷ ✶✳✸ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✶✳ ❇↔♥❣ ❝❤ú ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tỹ ụ tứ ❜↔♥❣ ❝❤ú ✶✳✶✳✹✳ ◆❣æ♥ ♥❣ú ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ tø ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✶✳ P❤➨♣ ♥❤➙♥ ❣❤➨♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✷✳ P❤➨♣ ❧➜② tø ♥❣÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✸✳ P❤➨♣ ❝❤✐❛ tø ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ ♥❣æ♥ ♥❣ú ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸✳✶✳ P❤➨♣ ❤ñ♣✱ ♣❤➨♣ ❣✐❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸✳✷✳ P❤➨♣ ❧➜② ♣❤➛♥ ❜ò ✶✳✸✳✸✳ P❤➨♣ ♥❤➙♥ ❣❤➨♣ ✶✳✸✳✹✳ P❤➨♣ ❧➦♣ ✶✳✸✳✺✳ P❤➨♣ ❧➜② ♥❣ỉ♥ ♥❣ú ♥❣÷đ❝ ✶✳✸✳✻✳ P❤➨♣ ❝❤✐❛ ♥❣æ♥ ♥❣ú ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷ ❱❿◆ P❍❸▼ ❱⑨ ◆●➷◆ ◆●Ú ❙■◆❍ ❇Ð■ ❱❿◆ P❍❸▼ ✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➠♥ ♣❤↕♠ ✷✳✷ ◆❣æ♥ ♥❣ú s✐♥❤ ❜ð✐ ✈➠♥ ♣❤↕♠ ✷✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✷ ✶✸ ✷✳✷✳✶✳ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷✳✷✳ ◆❣æ♥ ♥❣ú ❝õ❛ ✈➠♥ ♣❤↕♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷✳✸✳ ❱➠♥ ♣❤↕♠ t÷ì♥❣ ✤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ P❤➙♥ ❧♦↕✐ ✈➠♥ ♣❤↕♠ t❤❡♦ ❈❤♦♠s❦② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✸✳✶✳ ✶✽ ❱➠♥ ♣❤↕♠ ♥❣ú ❝➜✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ▼Ö❈ ▲Ö❈ ✹ ✷✳✸✳✷✳ ❱➠♥ ♣❤↕♠ ❝↔♠ ♥❣ú ❝↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸✳✸✳ ❱➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸✳✹✳ ❱➠♥ ♣❤↕♠ ❝❤➼♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✹ ❚➼♥❤ ✤â♥❣ ❝õ❛ ❧ỵ♣ ♥❣ỉ♥ ♥❣ú s✐♥❤ ❜ð✐ ✈➠♥ ♣❤↕♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✺ ▼ët sè ✈➼ ❞ö ✈➲ ✈➠♥ ♣❤↕♠ ✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❱❿◆ P❍❸▼ P❍■ ◆●Ú ❈❷◆❍ ✸✳✶ ✸✳✷ ✸✳✸ ✷✺ ❙✉② ❞➝♥ ♣❤✐ ♥❣ú ❝↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✶✳✶✳ ❈➙② s✉② ❞➝♥ ✤➛② ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✶✳✷✳ ❙ü ♥❤➟♣ ♥❤➡♥❣ tr♦♥❣ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✳ ✳ ✳ ✳ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ❇✐➳♥ ✤ê✐ ✈➠♥ ♣❤↕♠ ỳ ọ ỵ ổ ➼❝❤ ✸✳✷✳✷✳ ▲♦↕✐ ❜ä ❝→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷✳✸✳ ▲♦↕✐ ❜ä ❝→❝ q✉② t➢❝ ✤ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ❉↕♥❣ ❝❤✉➞♥ ❈❤♦♠s❦② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✸✳✶✳ ❱➠♥ ♣❤↕♠ ❝❤✉➞♥ ❝õ❛ ❈❤♦♠s❦② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✸✳✷✳ ✣÷❛ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✈➲ ❞↕♥❣ ❝❤✉➞♥ ❈❤♦♠s❦② ✸✸ ε✲q✉② t➢❝ s✐♥❤ ❑➌❚ ▲❯❾◆ ✸✹ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✸✺ ▲❮■ ◆➶■ ✣❺❯ ❚r♦♥❣ t❤í✐ ❜✉ê✐ ❝ỉ♥❣ ♥❣❤➺ t❤ỉ♥❣ t✐♥ ❜ò♥❣ ♥ê t❤➻ ♠→② t➼♥❤ ❧➔ ♥❤ú♥❣ t❤✐➳t ❜à ❦❤ỉ♥❣ t❤➸ t❤✐➳✉ tr♦♥❣ ✤í✐ sè♥❣ ❤➡♥❣ ♥❣➔② ❝↔ ❝ỉ♥❣ ✈✐➺❝ ❧➝♥ ❣✐↔✐ tr➼✳ ❱➟② ❧➔♠ t❤➳ ♥➔♦ ✤➸ ❝♦♥ ữớ õ t t ữủ ợ t t ởt tr ỳ ỵ tt ỡ s ỹ ♥➔② ✤â ❝❤➼♥❤ ❧➔ ❱➠♥ ♣❤↕♠ ✈➔ ♥❣æ♥ ♥❣ú ♣❤✐ ♥❣ú ❝↔♥❤✳ ❱➠♥ ♣❤↕♠ ♠ët ❝ỉ♥❣ ❝ư ✤➸ s↔♥ s✐♥❤ ♥❣ỉ♥ ♥❣ú✱ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ t❤ü❝ t➳ ✈➔ t t tr ổ ỷ ỵ ♥❣æ♥ ♥❣ú tü ♥❤✐➯♥✳ ◆❣æ♥ ♥❣ú ♣❤✐ ♥❣ú ❝↔♥❤ ❧➔ ♥❣ỉ♥ ♥❣ú ❦❤→ ❣➛♥ ✈ỵ✐ ♥❣ỉ♥ ♥❣ú tü ♥❤✐➯♥ ✈➔ ❝â ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ù♥❣ ❞ö♥❣ ①➙② ❞ü♥❣ ❝→❝ ♥❣ỉ♥ ♥❣ú ❧➟♣ tr➻♥❤ ✈➔ ❝❤÷ì♥❣ tr➻♥❤ ❞à❝❤✳ õ ỗ õ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✱ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ♥❣ỉ♥ ♥❣ú ❤➻♥❤ t❤ù❝ ❜❛♦ ỗ ỡ t tr tø ✈➔ tr➯♥ ♥❣ỉ♥ ♥❣ú✳ ❈❤÷ì♥❣ ✷✿ ❱➠♥ ♣❤↕♠ ✈➔ ♥❣æ♥ ♥❣ú s✐♥❤ ❜ð✐ ✈➠♥ ♣❤↕♠✱ tr➻♥❤ ❜➔② ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ✈➠♥ ♣❤↕♠ ✈➔ ♥❣æ♥ ♥❣ú s✐♥❤ ❜ð✐ ✈➠♥ ♣❤↕♠✳ ◆❣♦➔✐ r❛✱ ♣❤➛♥ ♥➔② ❝ô♥❣ ♣❤➙♥ ❧♦↕✐ r❛ ♠ët sè ✈➠♥ ♣❤↕♠ ✤➦❝ ❜✐➺t t❤❡♦ ❈❤♦♠s❦②✳ ❈❤÷ì♥❣ ✸✿ ❱➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤✱ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ ✈➲ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✈➔ ♠ët sè ❝→❝❤ ❜✐➳♥ ✤ê✐ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ♥❤➡♠ ❣✐↔♥ ❧÷đ❝ ♥â ✈➔ ✤÷❛ ♥â ✈➲ ♠ët tr♦♥❣ ♥❤ú♥❣ ❞↕♥❣ ❝❤✉➞♥ ✭❝ư t❤➸ ❧➔ ❝❤✉➞♥ ❈❤♦♠s❦②✮✳ ✺ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✶✳✶✳✶✳ ❇↔♥❣ ❝❤ú ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ▼ët t➟♣ ❤ú✉ ❤↕♥ ❦❤→❝ ré♥❣ ❝❤ú✳ ▼é✐ ♣❤➛♥ tû tr♦♥❣ Σ Σ ✤÷đ❝ ❣å✐ ❧➔ ởt ữủ ởt ỵ tỹ ỳ t ỗ ỵ tỹ Σ = {a, b, c , y, z} ✷✳ ❇↔♥❣ ❝❤ú ✤➸ s✐♥❤ ❝→❝ ①➙✉ ♥❤à ♣❤➙♥ ❧➔✿ ✸✳ ❇↔♥❣ ❝❤ú sè ▲❛ ▼➣✿ Σ = {0, 1} Σ = {I, V, X, L, C, D, M } ỵ tỹ ởt ỵ tỹ ởt tứ tr ỳ ỳ ỵ tỹ tr Σ ❧➔ ♠ët ❞➣② ✤÷đ❝ ✈✐➳t ❧✐➲♥ ♥❤❛✉✳ ❧➔ sè ỵ tỹ õ t tr ỵ || l (ω) ❳➙✉ ré♥❣ ❧➔ ①➙✉ ✤➦❝ ❜✐➺t ❦❤æ♥❣ ự ỵ tỹ rộ õ ỵ rộ ε✳ ✤÷đ❝ ❣å✐ ❧➔ ①➙✉ ❝♦♥ ❝õ❛ ①➙✉ ♠ët ❞➣② ỵ tr ✤÷đ❝ t↕♦ t❤➔♥❤ ❜ð✐ α✳ ❱➼ ❞ư ✶✳✷✳ ❙tr✐♥❣ ✶✳ ✧ ✧ ❧➔ ♠ët ①➙✉ tr➯♥ ❜↔♥❣ ❝❤ú l (ω) = ✧r✐♥❣✧ Σ = {a, b, c , y, z} ❧➔ ♠ët ①➙✉ ❝♦♥ ❝õ❛ ✻ ✧❙tr✐♥❣✧ ✳ ✈➔ ❝â ✤ë ❞➔✐ ❈❤÷ì♥❣ ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✼ ✷✳ α = accbaaab, β = aaabcba l (α) = 8✱ l (β) = ✸✳ ω = 0101011 (l (ω) = 7) ❧➔ ❤❛✐ ①➙✉ tr➯♥ ❜↔♥❣ ❝❤ú ❧➔ ♠ët ①➙✉ tr➯♥ ❜↔♥❣ ❝❤ú Σ = {0, 1} Σ = {a, b, c}✱ ❝â ✤ë ❞➔✐ ❧➔ ✼ ✶✳✶✳✸✳ ▲ô② t❤ø❛ ❝õ❛ ❜↔♥❣ ❝❤ú ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ Σ ữủ ỵ k k ợ k > ụ tứ ỳ ỵ tỹ tr Σ ❝â ✤ë ❞➔✐ ❜➡♥❣ ❧➔ t➟♣ t➜t ❝↔ k = t❤➻ Σ0 ❧➔ t➟♣ ❤ñ♣ ❝❤➾ ❝â ♠ët rộ = {} + ỵ ❤✐➺✉ Σ ✱ Σ ❧➛♥ ❧÷đt ❧➔ t➙♣ t➜t ❝↔ ỵ tỹ ỵ tỹ ❦❤→❝ ré♥❣ tr➯♥ Σ✳ ❉♦ ✤â✿ Σ∗ = Σ0 ∪ Σ1 ∪ Σ2 ∪ = Σ+ ∪ {ε} Σ+ = Σ1 ∪ Σ2 ∪ = Σ∗ − {ε} ❱ỵ✐ ❱➼ ❞ư ✶✳✸✳ ❱ỵ✐ Σ ❂ {x, y, z} t❤➻ Σ2 ré♥❣ ✈➔ {xx, xy, xz, yx, yy, yz, zx, zy, zz}✳ ❂ ✶✳✶✳✹✳ ◆❣æ♥ ♥❣ú ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ▼ët t➟♣ ❝♦♥ L ❝õ❛ t❤ù❝ ✭❤❛② ♥❣æ♥ ỳ tr ỳ ỵ ữủ ❧➔ ♠ët ♥❣æ♥ ♥❣ú ❤➻♥❤ Σ ❧➔ ♥❣æ♥ ♥❣ú ré♥❣✱ tù❝ ❧➔ ♥❣æ♥ ♥❣ú ❦❤æ♥❣ ❝❤ù❛ tø ♥➔♦ ❝↔✳ ❚❛ ❝➛♥ ♣❤➙♥ ❜✐➺t ♥❣ỉ♥ ♥❣ú ré♥❣ (∅) ✈ỵ✐ ♥❣ỉ♥ ♥❣ú ❝â ♠ët tø ré♥❣ {ε}✳ ◆❤÷ ✈➟② ♠ët ♥❣ỉ♥ ♥❣ú ❝â t❤➸ ❝â ❤ú✉ ❤↕♥ ❤♦➦❝ ✈æ ❤↕♥ tø✳ ❈❤➥♥❣ ❤↕♥✿ • Σ+ ✱ Σ∗ ❧➔ ❝→❝ ♥❣ỉ♥ ♥❣ú ✈ỉ ❤↕♥✳ ❳➨t ✈➲ ❝➜✉ tró❝ ✤↕✐ sè ❝â t❤➸ ❝❤ù♥❣ ữủ r k ợ k + ❧➔ ❝→❝ t➟♣ ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝✳ ❧➔ sè ❝ư t❤➸ ♥➔♦ ✤â ❧➔ ♥❣æ♥ ♥❣ú ❤ú✉ ❤↕♥✳ ✶✳✷ ❈→❝ ♣❤➨♣ t♦→♥ tr➯♥ tø ✶✳✷✳✶✳ P❤➨♣ ♥❤➙♥ ❣❤➨♣ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ α ❂ a1 a2 am ✈➔ tø β ❂ b1 b2 bn tr➯♥ ❜↔♥❣ ❝❤ú Σ ❧➔ tø γ ❂ a1 a2 am b1 b2 bn tr➯♥ ❜↔♥❣ ❝❤ú Σ✳ ❚➼❝❤ ❣❤➨♣ ✭❤❛② ♥❤➙♥ ❣❤➨♣✮ ❝õ❛ ❤❛✐ tø ❈❤÷ì♥❣ ỵ ữợ = = ợ ❂ α.β ∀α ✽ γ ✭❤❛② ❂ αβ ✮✳ ❚❛ t❤➜② ♣❤➨♣ ♥❤➙♥ ❣❤➨♣ ❝â t➼♥❤ ❦➳t ❤ñ♣✳ ❚ù❝ ❧➔ ✈ỵ✐ ∀α, β, γ t❛ ✤➲✉ ❝â✿ (αβ) γ = α (βγ) ❱➼ ❞ö ✶✳✹✳ ✶✳ α = 01101, β = 110 ✷✳ α = aab, β = bccab t❤➻ t❤➻ αβ = 01101110 αβ = aabbccab ✶✳✷✳✷✳ P❤➨♣ ❧➜② tø ♥❣÷đ❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ am am−1 a2 a1 R ữợ sỷ a1 a2 am ❦❤→❝ ré♥❣ tr➯♥ ❜↔♥❣ ❝❤ú ✤÷đ❝ ❣å✐ ❧➔ tø ♥❣÷đ❝ ✭❤❛② tø s♦✐ ❣÷ì♥❣✮ ❝õ❛ tø εR ứ ỵ õ t➼♥❤ ❝❤➜t s❛✉✿ • ωR R =ω • (αβ)R = β R αR • |ω|R = |ω| ❱➼ ❞ư ✶✳✺✳ α = aabc, β = abba✳ ❑❤✐ ✤â✿ α = cbaa, β = abba✱ (αβ)R = (aabcabba)R = abbacbaa = β R αR R ❈❤♦ R ✶✳✷✳✸✳ P❤➨♣ ❝❤✐❛ tø P❤➨♣ ❝❤✐❛ tø ❧➔ ♣❤➨♣ t♦→♥ ♥❣➢t ❜ä ♣❤➛♥ ✤➛✉ ❤❛② ♣❤➛♥ ❝✉è✐ ❝õ❛ ♠ët tø✳ ❚❛ ❝â ✤à♥❤ ♥❣❤➽❛ s❛✉✿ ✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❈❤♦ α, β, γ ❛✮ P❤➨♣ ❝❤✐❛ tr→✐ ❝õ❛ tø ❜✮ P❤➨♣ ❝❤✐❛ ♣❤↔✐ ❝õ❛ tø ❧➔ ❝→❝ tø tr➯♥ Σ✳ α ❝❤♦ tø β ✭❤❛② t❤÷ì♥❣ ❜➯♥ tr→✐ ❝õ❛ α ✈➔ β ✮ ❧➔ ♣❤➛♥ ❝á♥ ❧↕✐ ❝õ❛ tø α s❛✉ ❦❤✐ ♥❣➢t ❜ä tr tứ ữủ ỵ \ ỏ ữủ ỵ ❧➔ α ❝❤♦ tø γ ✭❤❛② t❤÷ì♥❣ ❜➯♥ ♣❤↔✐ ❝õ❛ α ✈➔ γ ✮ tø α s❛✉ ❦❤✐ ♥❣➢t ❜ä ♣❤➛♥ ❝✉è✐ γ tr♦♥❣ tø α ✈➔ α/γ ✳ ❈❤÷ì♥❣ ✷✳ ✕ ❱❿◆ P❍❸▼ ❱⑨ ◆●➷◆ ◆●Ú ❙■◆❍ ❇Ð■ ❱❿◆ P❍❸▼ ◆➳✉ ✷✷ ∆1 ∩∆2 = ∅ t❤➻ t❛ ✤ê✐ ỵ = ∅✳ ✕ ❇✐➳♥ ✤ê✐ G1 ✱ G2 t❤➔♥❤ ❤❛✐ ✈➠♥ ổ ự ỵ ỡ tr→✐✳ ✕ G❂< Σ, ∆, S, P > tr♦♥❣ ✤â✿ Σ = Σ1 ∪ Σ2 ✱ ∆ = ∆1 ∪ ∆2 ∪ {S}✱ P = P1 ∪ P2 ∪ {S → S1 , S → S2 } ❳→❝ ✤à♥❤ ❉♦ ✤â ✈➠♥ ♣❤↕♠ G ❧➔ ✤â♥❣ ✤è✐ ✈ỵ✐ ♣❤➨♣ ủ ú ỵ ữớ t ự ữủ r ▲ỵ♣ ♥❣ỉ♥ ♥❣ú s✐♥❤ ❜ð✐ ✈➠♥ ♣❤↕♠ ❝ơ♥❣ ✤â♥❣ ✤è✐ ✈ỵ✐ ❝→❝ ♣❤➨♣ t♦→♥ tr➯♥ ♥❣ỉ♥ ♥❣ú✿ ♣❤➨♣ ❧➦♣✱ ❧➦♣ ❝➢t✱ ♣❤➨♣ ❝❤✐❛ tr→✐ ✈➔ ❝❤✐❛ ♣❤↔✐✳ ✷✳ ▲ỵ♣ ♥❣ỉ♥ ♥❣ú s✐♥❤ ❜ð✐ ✈➠♥ ♣❤↕♠ ❦❤ỉ♥❣ ✤â♥❣ ✤è✐ ✈ỵ✐ ♣❤➨♣ trø ✈➔ ♣❤➨♣ ❧➜② ♣❤➛♥ ❜ò ♥❣ỉ♥ ♥❣ú✳ ❍➺ q✉↔ ✷✳✶✳ ◆➳✉ L1 ✈➔ L2 ❧➔ ❤❛✐ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② ✭t÷ì♥❣ ù♥❣ ♣❤✐ ♥❣ú ❝↔♥❤✱ ❝↔♠ ♥❣ú ❝↔♥❤✮ t❤➻ L1 ∪ L2 ❝ơ♥❣ ❧➔ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉② ✭t÷ì♥❣ ù♥❣ ♣❤✐ ♥❣ú ❝↔♥❤✱ ❝↔♠ ♥❣ú ❝↔♥❤✮✳ ❍➺ q✉↔ ✷✳✷✳ ◆➳✉ L1 ✈➔ L2 ❧➔ ❤❛✐ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② ✭t÷ì♥❣ ù♥❣ ♣❤✐ ♥❣ú ❝↔♥❤✱ ❝↔♠ ♥❣ú ❝↔♥❤✮ t❤➻ L1 L2 ❝ơ♥❣ ❧➔ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉② ✭t÷ì♥❣ ù♥❣ ♣❤✐ ♥❣ú ❝↔♥❤✱ ❝↔♠ ♥❣ú ❝↔♥❤✮✳ ❍➺ q✉↔ ✷✳✸✳ ◆➳✉ L ❧➔ ♥❣æ♥ ♥❣ú ❝❤➼♥❤ q✉② t❤➻ ♥❣æ♥ ♥❣ú ❧➦♣ L∗ ❝õ❛ L ❝ơ♥❣ ❧➔ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉②✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ❧ỵ♣ ❝→❝ ♥❣ỉ♥ ♥❣ú ❝❤➼♥❤ q✉② ✤â♥❣ ✤è✐ ✈ỵ✐ ♣❤➨♣ t♦→♥ ❧➦♣✳ ✷✳✺ ▼ët sè ✈➼ ❞ö ✈➲ ✈➠♥ ♣❤↕♠ ❱➼ ❞ư ✷✳✾✳ ❍➣② ①→❝ ✤à♥❤ ♥❣ỉ♥ ♥❣ú ❞♦ ✈➠♥ ♣❤↕♠ s❛✉ ✤➙② s✐♥❤ r❛✿ G❂< Σ, ∆, S, P >✱ ✈ỵ✐✿ Σ = {a, b}✱ ∆ = {S, A, B} P ❂{S → aA, S → bB, A → aA, A → ε, B → bB, B → ε} ❇➔✐ ❧➔♠✿ ❚❛ ✤→♥❤ sè ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ P t❤❡♦ t❤ù tü tø tr→✐ ❈❤♦ ✈➠♥ ♣❤↕♠ ♣❤↔✐ ❧➔ ✭✶✮✱ ✭✷✮✱ ✭✸✮✱ ✭✹✮✱ ✭✺✮✱ ✭✻✮✳ q✉❛ ❈❤÷ì♥❣ ✷✳ ❱❿◆ P❍❸▼ ❱⑨ ◆●➷◆ ◆●Ú ❙■◆❍ ❇Ð■ ❱❿◆ P❍❸▼ ❳➨t t➜t ❝↔ ♥❤ú♥❣ ❞➝♥ ①✉➜t ✤➛② ✤õ ❝â t❤➸ ❝â tr♦♥❣ ✈➠♥ ♣❤↕♠ D = (S, ω1 , , ωk )✳❚r♦♥❣ P ❝❤➾ ❝â ✷ q✉② t➢❝ s✐♥❤ ❝â ✈➳ tr→✐ ❧➔ ✷✸ G ❝â ❞↕♥❣ S ❧➔ ✭✶✮ ✈➔ ✭✷✮✳ ◆➳✉ D ❜➢t ✤➛✉ ❜ð✐ ✭✶✮ tù❝ ❧➔ ω1 = aA✱ t✐➳♣ t❤❡♦ ❝❤➾ ❝â t❤➸ →♣ ❞ö♥❣ q✉② t➢❝ s✐♥❤ ✭✸✮ ❤♦➦❝ ✭✹✮✳ ◗✉② t➢❝ s✐♥❤ ✭✸✮ s s t ởt ỵ tỹ t a ỳ ♥❣✉②➯♥ A✱ →♣ ❞ö♥❣ ❜❛♦ ♥❤✐➯✉ ❧➛♥ q✉② t➢❝ s✐♥❤ ✭✸✮ s➩ s✐♥❤ r❛ ❜➜② ♥❤✐➯✉ a✳ ❑❤✐ ✤â q✉② t➢❝ s✐♥❤ ✭✹✮ s➩ tr✐➺t t✐➯✉ A ✈➔ ❦➳t t❤ó❝ ❞➝♥ ①✉➜t ✤➛② ✤õ✳ ❱➟② ♥➳✉ D →♣ ❞ö♥❣ ✭✶✮ ✤➛✉ t✐➯♥ t❤➻ s➩ s✐♥❤ r❛ tø ❝â ❞↕♥❣ aa a ợ k ỵ tỹ a k > ❧✉➟♥ t÷ì♥❣ tü✱ ♥➳✉ D →♣ ❞ư♥❣ ✭✷✮ ✤➛✉ t✐➯♥ t❤➻ s➩ s✐♥❤ r❛ tø ❝â ❞↕♥❣ bb b ợ k ỵ tỹ b k > ❤❛✐ ❞↕♥❣ tø ♥➯✉ tr➯♥✱ ❝→❝ ❞➝♥ ①✉➜t ✤➛② ✤õ tr♦♥❣ G ❦❤ỉ♥❣ s✐♥❤ t❤ó❝ r❛ ❞↕♥❣ tø ♥➔♦ ❦❤→❝✳ ❱➟② L (G) = {am , bn | m > 0, n > 0} ❱➼ ❞ö ✷✳✶✵✳ ❍➣② t➻♠ ✈➠♥ ♣❤↕♠ s✐♥❤ r❛ ♥❣æ♥ ♥❣ú L❂{am bn | m ≥ n ≥ 0} ❇➔✐ ❧➔♠✿ • P❤➙♥ t➼❝❤✿ ❚❛ t❤➜② ♥❣æ♥ ♥❣ú ✤➣ ❝❤♦ ❝❤➾ ❝â ♠ët ❞↕♥❣ tø ❜➢t ✤➛✉ ❧➔ a✱ ♠ët ❞➣② t✐➳♣ t❤❡♦ ❧➔ ởt b ợ số ỵ tỹ a ❦❤ỉ♥❣ ➼t b ✈➔ sè ❧÷đ♥❣ b ❝â t❤➸ ❧➔ ✵✳ ❱➻ ✈➟② tr♦♥❣ q✉→ tr➻♥❤ t↕♦ r❛ ❞↕♥❣ ♥➔② ❝➛♥ ✤➸ sè ❧÷đ♥❣ a ❦❤ỉ♥❣ ✤÷đ❝ ➼t ❤ì♥ sè ữủ b ỡ số ỵ tỹ tứ õ t s t s r ỗ tớ a ✈➔ ◗✉② t➢❝ s✐♥❤ ❝❤➾ s✐♥❤ r✐➯♥❣ r❛ ❱➻ sè ❧÷đ♥❣ b b a S → aSb ð ❤❛✐ ♣❤➼❛ ❧➔ ♠➔ ❦❤æ♥❣ ❝â ❝â t❤➸ ❧➔ ✵ ✈➔ ❦❤✐ ✤â sè ❧÷đ♥❣ a b ❧➔ S → aS ❝ơ♥❣ ❝â t❤➸ ❧➔ ✵ tù❝ ❧➔ tø ré♥❣ ❝ô♥❣ t❤✉ë❝ ♥❣æ♥ ♥❣ú ✤➣ ❝❤♦✱ ✈➟② ♣❤↔✐ ❝â q✉② t➢❝ s✐♥❤ r❛ tø ré♥❣ S→ε ✤➙② ❝ô♥❣ ❝❤➼♥❤ ❧➔ q✉② t➢❝ s✐♥❤ ❦➳t t❤ó❝✳ ◆❤÷ ✈➟② ❝❤➾ ❝➛♥ ❜❛ q✉② t➢❝ s✐♥❤ ♥➔② ❝â t❤➸ t↕♦ r❛ t➜t ❝↔ ❝→❝ tø ❝õ❛ ♥❣ỉ♥ ♥❣ú ✤➣ ❝❤♦ ✈➔ ❝❤ó♥❣ ❦❤ỉ♥❣ t↕♦ r❛ tø ♥❣♦↕✐ ❧❛✐ ♥➔♦✳ ❱➟② ✈➠♥ ♣❤↕♠ ❝➛♥ t➻♠ ❧➔✿ G❂< Σ, ∆, S, P > ✈ỵ✐✿ Σ = {a, b}✱ ∆ = {S}✱ P = {S → aSb, S → aS, S → ε} ❱➼ ❞ö ✷✳✶✶✳ G❂< Σ, ∆, S, P > ✈ỵ✐ Σ = {a, b}✱ ∆ = {S, A}✱ P = {S → Sa, S → A, A → aAb, A → ab}✳ ❳→❝ ✤à♥❤ ♥❣ú ❝õ❛ G✳ ❈❤♦ ✈➠♥ ♣❤↕♠ ♥❣ỉ♥ ❈❤÷ì♥❣ ✷✳ ❱❿◆ P❍❸▼ ❱⑨ ◆●➷◆ ◆●Ú ❙■◆❍ ❇Ð■ ❱❿◆ P❍❸▼ G ❝❤➾ ❝â ❞↕♥❣ ♥❤÷ s❛✉✿ D = S, Sa, , Sa , Aa , aAba , , am Abm an , am+1 bm+1 an ✈ỵ✐ m ≥ 0, n ≥ 0✳ m m n ❱➟② ♥❣æ♥ ♥❣ú s✐♥❤ ❜ð✐ G ❧➔✿ L (G) = {a b a | n > 0, m ≥ 0} ❇➔✐ ❧➔♠✿ ❉➝♥ ①✉➜t ✤➛② ✤õ tr♦♥❣ n n n ✷✹ ❈❤÷ì♥❣ ✸ ❱❿◆ P❍❸▼ P❍■ ◆●Ú ❈❷◆❍ ✸✳✶ ❙✉② ❞➝♥ ♣❤✐ ♥❣ú ❝↔♥❤ ✸✳✶✳✶✳ ❈➙② s✉② ❞➝♥ ✤➛② ✤õ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳ ❈❤♦ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ s✉② ❞➝♥ ✤➛② ✤õ tr♦♥❣ ✈➠♥ ♣❤↕♠ G❂< Σ, ∆, S, P >✳ ❈➙② G ❧➔ ♠ët ỗ t ỳ õ ữợ ổ õ tr ✈➔ t❤ä❛ ♠➣♥ ❜è♥ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ ▼é✐ ✤➾♥❤ ữủ ởt ỵ tr♦♥❣ t➟♣ Σ∪ ∆ ∪ {ε}✳ ●è❝ ❝õ❛ ❝➙② ✤÷đ❝ ❣→♥ ♥❤➣♥ ❧➔ S✳ ✷✳ ▼é✐ ✤➾♥❤ tr♦♥❣ ✤÷đ❝ ❣→♥ ởt ỵ õ tr ▼é✐ ✤➾♥❤ ♥❣♦➔✐ ✭❧→ ❝õ❛ ❝➙②✮ ✤÷đ❝ ❣→♥ ♥❤➣♥ ❧➔ ởt ỵ tr t ộ m ✤÷đ❝ ❣→♥ ♥❤➣♥ ❧➔ ❝→❝ ❝♦♥ ❝õ❛ ✤➾♥❤ m A ∈ ∆✱ ❝á♥ ❝→❝ ✤➾♥❤ n1 , n2 , , nk ❧➔ t❤❡♦ t❤ù tü tø tr→✐ s❛♥❣ ♣❤↔✐ ✈➔ ✤÷đ❝ ❣→♥ ♥❤➣♥ B1 , B2 , , Bk t÷ì♥❣ ù♥❣ t❤➻ A → B1 B2 Bk ❧➔ ♠ët q✉② t➢❝ s✐♥❤ tr♦♥❣ P ❝õ❛ ✈➠♥ ♣❤↕♠ G✳ ◆➳✉ ✤å❝ t➜t ❝↔ ❝→❝ ♥❤➣♥ ❞➣♥ ð ❝→❝ ❧→ t❤❡♦ t❤ù tü tø tr→✐ s❛♥❣ ♣❤↔✐ ✱ t❛ s➩ ♥❤➟♥ ✤÷đ❝ ♠ët tø ♥➔♦ ✤â✳ ❚ø ✤â s➩ ❧➔ ♠ët ♣❤➛♥ tû tr♦♥❣ ❣å✐ ❧➔ ❦➳t q✉↔ ❝õ❛ ❝➙② s✉② ❞➝♥ tr♦♥❣ ❱➼ ❞ư ✸✳✶✳ L (G) ✈➔ ✤÷đ❝ G✳ G❂< Σ, ∆, S, P > ✈ỵ✐ Σ = {a, b}✱ ∆ = {S, A}✱ P = {S → Sa, S → Aa, A → aAb, A → ab} ▼ët ❝➙② s✉② ❞➝♥ ✤➛② ✤õ ❝õ❛ G ❝❤♦ ❦➳t q✉↔ ❧➔ ①➙✉ aaabbbaa ❝â ❞↕♥❣ ❈❤♦ ✈➠♥ ♣❤↕♠ ♥❤÷ s❛✉✿ ✷✺ ❈❤÷ì♥❣ ✸✳ ❱❿◆ P❍❸▼ P❍■ ◆●Ú ❈❷◆❍ ✷✻ ✸✳✶✳✷✳ ❙ü ♥❤➟♣ ♥❤➡♥❣ tr♦♥❣ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳ G❂< Σ, ∆, S, P >✳ ❱➠♥ ♣❤↕♠ G tỗ t ♠ët ①➙✉ ω ❧➔ ❦➳t q✉↔ ❝õ❛ ❤❛✐ ❝➙② s✉② ❞➝♥ ❦❤→❝ ♥❤❛✉ tr♦♥❣ G✳ ❈❤♦ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ◆❣÷đ❝ ❧↕✐✱ ✈➠♥ ♣❤↕♠ G ❧➔ ❦❤ỉ♥❣ ♥❤➟♣ ♥❤➡♥❣ ❤❛② ✤ì♥ ♥❣❤➽❛✳ ▼ët ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✤÷đ❝ ỷ ổ tỗ t ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✤ì♥ ♥❣❤➽❛ ♥➔♦ t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ ♥â✳ ◆❣ỉ♥ ♥❣ú ❞♦ ✈➠♥ ♣❤↕♠ G s✐♥❤ r❛ ❣å✐ ❧➔ ♥❣æ♥ ♥❣ú ♥❤➟♣ ♥❤➡♥❣ ♥➳✉ G ❧➔ ✈➠♥ ♣❤↕♠ ♥❤➟♣ ♥❤➡♥❣✳ ❱➼ ❞ö ✸✳✷✳ G❂< Σ, ∆, S, P > ✈ỵ✐✿ Σ = {+, ∗, a}✱ ∆ = {S}✱ P = {S → S + S, S → S ∗ S, S → a} ∗ ❳➨t ①➙✉ ω = a + a a ❈â ❤❛✐ ❝➙② s✉② ❞➝♥ ✤➛② ✤õ ❦❤→❝ ♥❤❛✉ ❝õ❛ G ✤➲✉ ❝❤♦ ❦➳t q✉↔ ❧➔ ①➙✉ ω : ❈❤♦ ✈➠♥ ♣❤↕♠ ❈❤÷ì♥❣ ✸✳ ❱❿◆ P❍❸▼ P❍■ ◆●Ú ❈❷◆❍ ❱➟② ✈➠♥ ♣❤↕♠ G ✷✼ ❧➔ ♥❤➟♣ ♥❤➡♥❣✳ ✸✳✷ ❇✐➳♥ ✤ê✐ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ❑❤✐ sû ❞ö♥❣ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✤➸ ♣❤➙♥ t➼❝❤ ❝ó ♣❤→♣✱ ♥❤➟♥ ❜✐➳t ❝→❝ tø ❝õ❛ ♥❣ỉ♥ ♥❣ú✱ ❧♦↕✐ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ♥❤➟♣ ♥❤➡♥❣ ❣➙② r t ổ t tr q tr ỷ ỵ ❦➨♦ t❤❡♦ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ❦❤ỉ♥❣ ❧÷í♥❣ ❤➳t ✤÷đ❝✳ ❱➻ ✈➟② ♣❤↔✐ ❦❤û t➼♥❤ ♥❤➟♣ ♥❤➡♥❣ ❝õ❛ ✈➠♥ ♣❤↕♠ ♣❤✐ ỳ ọ ỵ ổ G< , , S, P > ỵ ❤✐➺✉ X ∈ (Σ ∪ ∆) ✤÷đ❝ ❣å✐ ❧➔ ❝â tỗ t s S X ω ✱ tr♦♥❣ ✤â α, β ❧➔ ❝→❝ ①➙✉ tr✉♥❣ ❣✐❛♥ t❤✉ë❝ (Σ ∪ ∆) ✱ ω ∈ Σ ✳ ỵ X ổ tọ tr t ỵ tứ ỵ ỳ tứ ỗ ỵ ổ s ỵ ổ ữủ X ữủ ỵ ổ s ổ tỗ t↕✐ s✉② ❞➝♥ X ω ✱ ω ∈ Σ∗ ỵ X ( ) ữủ ỵ ổ ữủ ổ tỗ t s ❞➝♥ ❝â ❞↕♥❣ S αXβ tr♦♥❣ ✤â α, β ❧➔ tở ( ) ỵ t t ọ ỵ ổ s t tt G< Σ, ∆, S, P >✳ ❱➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ G ❂< Σ, ∆ , S, P > s✐♥❤ ✈➔ L (G ) = L (G) ❱➠♥ ♣❤↕♠ ♣❤✐ ỳ ổ ổ ự ỵ ữỡ P P ữợ = ❳➙② ❞ü♥❣ ❳✉➜t ♣❤→t ❝❤♦ ❳➨t ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ ❤✐➺✉ A ✈➔♦ ✷✽ P✿ ◆➳✉ A→α ✈ỵ✐ α t s ỵ A X1 X2 Xk ♠➔ Xi ∈ Σ ❤♦➦❝ Xi ∈ ∆ ✈ỵ✐ (i = 1, 2, , k) t❤➻ ❜ê s✉♥❣ t ỵ A q tr➻♥❤ ♥➔② ❝❤♦ ✤➳♥ ❦❤✐ ①➨t ❤➳t ❝→❝ q✉② t➢❝ s P ổ s t ỵ ♥➔♦ ♠ỵ✐ ✈➔♦ ∆ ✳ ◗✉→ tr➻♥❤ ♥➔② ❝❤➢❝ ❝❤➢♥ ♣❤↔✐ ❞ø♥❣ ❧↕✐ ✈➻ P ❳➨t ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ P✿ ◆➳✉ P✿ ◆➳✉ ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ P P = ữợ ỹ t t ❳➨t ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ A → α ❝â ỵ ( ) t❤➻ q✉② t➢❝ s✐♥❤ ♥➔② ✤÷đ❝ ❜ê s✉♥❣ ✈➔♦ P ✳ ▲➦♣ ❧↕✐ q✉→ ♥➔② ❝❤♦ ✤➳♥ ❦❤✐ ①➨t ❤➳t ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ P ✈➔ ❦❤æ♥❣ ❜ê s✉♥❣ q✉② t➢❝ s✐♥❤ ♥➔♦ ✈➔♦ P ✳ ◗✉→ tr➻♥❤ ♥➔② ❝❤➢❝ ❝❤➢♥ ♣❤↔✐ ❞ø♥❣ ❧↕✐ ✈➻ ✤➲✉ t❤✉ë❝ tr➻♥❤ t❤➯♠ P ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ ❉➵ ❞➔♥❣ ❝â t❤➸ t r ỳ ữỡ ợ ♣❤↕♠ ✤➣ ❝❤♦ G ✈➔ tr♦♥❣ G G t↕♦ ✤÷đ❝ tữỡ ổ ự ỵ ổ s t t ọ ỵ ổ ữủ t tt G< , ∆, S, P > ❱➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ G ❂< Σ , ∆ , S, P > ❦❤æ♥❣ ❝❤ù❛ ỵ ổ ữủ L (G ) = L (G) ữợ ỹ t ♣❤→t ❝❤♦ Σ = ∅ ✈➔ ∆ = {S} ∗ ❳➨t ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ P ✿ ◆➳✉ A → α ✈ỵ✐ A ∈ ∆ ✈➔ α ∈ (Σ ) t s t ỵ ổ ❦➳t t❤ó❝ ❝❤ù❛ tr♦♥❣ α ✈➔♦ ∆ ✈➔ ❜ê s✉♥❣ t ỵ t tú ự tr ✳ ▲➦♣ ❧↕✐ q✉→ tr➻♥❤ ♥➔② ❝❤♦ ✤➳♥ ❦❤✐ ①➨t ❤➳t ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ P ✈➔ ❦❤æ♥❣ ❜ê s t ỵ ợ ✳ ◗✉→ tr➻♥❤ ♥➔② ❝❤➢❝ ❝❤➢♥ ♣❤↔✐ ❞ø♥❣ ❧↕✐ ✈➻ P ❧➔ t➟♣ ❤ú✉ ❱➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ❤↕♥✳ P P = ữợ ỹ t t ❳➨t ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ ✤➲✉ t❤✉ë❝ P✿ ◆➳✉ A õ ỵ ✈➳ (Σ ∪ ∆ ) t❤➻ q✉② t➢❝ s✐♥❤ ♥➔② ✤÷đ❝ ❜ê s✉♥❣ ✈➔♦ P ✳ ▲➦♣ ❧↕✐ q✉→ ❈❤÷ì♥❣ ✸✳ ❱❿◆ P❍❸▼ P❍■ ◆●Ú ❈❷◆❍ ✷✾ tr➻♥❤ ♥➔② ❝❤♦ ✤➳♥ ❦❤✐ ①➨t ❤➳t ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ t❤➯♠ q✉② t➢❝ s✐♥❤ ♥➔♦ ✈➔♦ P P P ✈➔ ❦❤æ♥❣ ❜ê s✉♥❣ ✳ ◗✉→ tr➻♥❤ ♥➔② ❝❤➢❝ ❝❤➢♥ ♣❤↔✐ ❦➳t t❤ó❝ ✈➻ ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ ❉➵ ❞➔♥❣ t❤➜② r➡♥❣ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✈ỵ✐ ✈➠♥ ♣❤↕♠ G ✤➣ ❝❤♦ ✈➔ tr♦♥❣ G G t↕♦ ✤÷đ❝ t÷ì♥❣ ✤÷ì♥❣ ❦❤ỉ♥❣ ự ỵ ổ ữủ t ứ t❤✉➟t t♦→♥ tr➯♥ t❛ ❝â ❦➳t q✉↔✿ ❱ỵ✐ ♠é✐ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ G❂< Σ, ∆, S, P > ổ tỗ t ỳ tữỡ ữỡ ổ ự ỵ tứ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ t÷ì♥❣ ✤÷ì♥❣ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ t ữợ ữ s ữợ ♣❤✐ ♥❣ú ❝↔♥❤ G G✱ G ❦❤ỉ♥❣ G✱G ❦❤ỉ♥❣ t÷ì♥❣ ữỡ ợ ự ỵ ổ s ữợ ỳ G tữỡ ữỡ ợ ự ỵ ổ ữủ G ❧➔ ✈➠♥ ♣❤↕♠ ❝➛♥ t➻♠✳ ❱➼ ❞ö ✸✳✸✳ G❂< Σ, ∆, S, P > ✈ỵ✐✿ Σ = {a}✱ ∆ = {S, A, B}✱ P = {S → AB, S → a, A → a} ✈➠♥ ♣❤↕♠ s✐♥❤ r❛ ♥❣æ♥ ♥❣ú L (G) = {a} t ỗ ởt a ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ G ♥❤➟♥ ✤÷đ❝ tø G ❜➡♥❣ ❝→❝❤ →♣ ❞ö♥❣ ❝→❝ ❦➳t ❈❤♦ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú q ữ tr ổ ự ỵ ❤✐➺✉ t❤ø❛ ❝➛♥ t➻♠ ❝â ❞↕♥❣ ♥❤÷ G ❂< Σ , ∆ , S, P > ✈ỵ✐ Σ = {a}✱ ∆ = {S}✱ P = {S → a} ❱➟② L (G ) = L (G) = {a} s❛✉✿ ú ỵ rt tỹ tt t ọ ổ s rỗ tt t ọ ổ ✤÷đ❝✮ ♥❤÷ tr➯♥ ❧➔ ❝➛♥ t❤✐➳t✳ ✣↔♦ ❧↕✐ ❝â t❤➸ ❞➝♥ tỵ✐ ❦➳t q✉↔ s❛✐✳ ✸✳✷✳✷✳ ▲♦↕✐ ❜ä ❝→❝ ε✲q✉② t➢❝ s✐♥❤ ✣à♥❤ ♥❣❤➽❛ ✸✳✹✳ tr♦♥❣ P tr♦♥❣ ❝â q✉② t➢❝ s✐♥❤ ε ∈ L (G) G ♣❤↔✐ ❝❤ù❛ ◆➳✉ G❂< Σ, ∆, S, P >✱ ♥➳✉ t❛ ♥â✐ G ❝â ε✲q✉② t➢❝✳ ε✲q✉② t➢❝ ❦❤ä✐ G ✭➼t ♥❤➜t ❈❤♦ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ A→ε ✈➔ A→∆ t❤➻ t❛ ❦❤æ♥❣ t❤➸ ❧♦↕✐ ❤➳t ♠å✐ q✉② t➢❝ s✐♥❤ S → ε✮✳ ❈❤÷ì♥❣ ✸✳ ◆➳✉ ❱❿◆ P❍❸▼ P❍■ ◆●Ú ❈❷◆❍ ε∈ / L (G) ✸✵ t❛ ❝â t❤➸ ❧♦↕✐ ❜ä t♦➔♥ ởt ợ tữỡ ữỡ ợ q t ❦❤ä✐ L (G) ✤➸ ✤÷đ❝ G✳ ❚❤✉➟t t♦→♥ ❧♦↕✐ ❜ä ε✲q✉② t➢❝✿ ■♥♣✉t✿ ❖✉t♣✉t✿ G❂< Σ, ∆, S, P > ❱➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ G ❂< Σ, ∆, S, P > ❦❤æ♥❣ ❝❤ù❛ ε✲q✉② t➢❝ ✈➔ L (G ) = L (G) \ {} ữợ A → ε ❧➔ ε✲q✉② t➢❝ tr♦♥❣ G t❤➻ A ữủ ỵ trt ỳ t ữợ Xi1 Xi2 , , Xik A → X1 X2 Xn ❧➔ q✉② t➢❝ s✐♥❤ tr♦♥❣ ỵ trt t t s P G ♠➔ tr♦♥❣ ✤â t➜t ❝↔ ❝→❝ q✉② A → X1 X2 Xn tr♦♥❣ ✤â ♠ët ❤♦➦❝ ♠ët sè ỵ trt t Xi1 Xi2 , , Xik ữủ t❤❛② ❜ð✐ ε ✭♥➳✉ q✉② t➢❝ s✐♥❤ t↕♦ t❤➔♥❤ ❦❤æ♥❣ q t ữợ s P tt ❝↔ ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ P ♠➔ ❦❤æ♥❣ ♣❤↔✐ ε✲q✉② t➢❝✳ t➢❝ s✐♥❤ ❝â ❞↕♥❣ ❱➼ ❞ö ✸✳✹✳ G❂< Σ, ∆, S, P > ✈ỵ✐✿ Σ = {a, b}✱ ∆ = {S, A, B} P = {A → ε, A → a, B → ε, B → b, S → ABA, S → aS} ❚❛ ①➙② ❞ü♥❣ ✈➠♥ ♣❤↕♠ G ❦❤æ♥❣ ❝❤ù❛ ε✲q✉② t➢❝ ✈➔ L (G ) = L (G) \ {} t A, B ỵ ❤✐➺✉ tr✐➺t t✐➯✉ ♥➯♥ ❧♦↕✐ ❜ä ε✲q✉② t➢❝ A → ε ✈➔ B → ε ❦❤ä✐ P ✈➔ ❜ê s✉♥❣ ✈➔♦ P ❝→❝ q✉② t➢❝ s✐♥❤ r❛ tø ♣❤➨♣ t❤➳ S → ABA ❧➔✿ ❈❤♦ S → AB, S → BA, S → AA, S → B ✳ ❑➳t ❤ñ♣ ✈ỵ✐ ❝→❝ q✉② t➢❝ s✐♥❤ ❦❤→❝ ε✲q✉② t➢❝ tr♦♥❣ P t❛ ❝â✿ P = {A → a, S → ABA, S → AB, B → b, S → BA, S → B, S → AA, S → aS} ❱➟② G ❂< Σ, ∆, S, P > ❧➔ ✈➠♥ ♣❤↕♠ ❝➛♥ t➻♠✳ ✸✳✷✳✸✳ ▲♦↕✐ ❜ä ❝→❝ q✉② t➢❝ ✤ì♥ ✣à♥❤ ♥❣❤➽❛ ✸✳✺✳ t➢❝ s✐♥❤ tr♦♥❣ P ❈❤♦ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ❝â ❞↕♥❣ A→B ✈ỵ✐ A, B ∈ ∆ G❂< Σ, ∆, S, P >✳ ◗✉② ✤÷đ❝ ❣å✐ ❧➔ q✉② t➢❝ ✤ì♥✳ ❈❤÷ì♥❣ ✸✳ ❱❿◆ P❍❸▼ P❍■ ◆●Ú ❈❷◆❍ ✸✶ ❚❤✉➟t t♦→♥ ❧♦↕✐ ❜ä q✉② t➢❝ ✤ì♥✿ ■♥♣✉t✿ ❖✉t♣✉t✿ G❂< Σ, ∆, S, P > ❝↔♥❤ G ❂< Σ, ∆, S, P > ❱➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ❱➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú L (G ) = L (G) ✶✿ ◆➳✉ A → B ❧➔ q✉② ❦❤æ♥❣ ❝❤ù❛ q✉② t ỡ ữợ t ỡ t ữủ ỵ ỡ tữỡ ự ữợ X X1 X2 Xn ◆➳✉ Xi1 , Xi2 , , Xik A ữủ ỵ ỡ B A ởt q t ỵ ỡ t t ❜ê s✉♥❣ ✈➔♦ s✐♥❤ P tr♦♥❣ P ✈➔ t➜t ❝↔ ❝→❝ q✉② t➢❝ s✐♥❤ ❤➻♥❤ t❤➔♥❤ tø q✉② t➢❝ s✐♥❤ tr t ởt ởt số ỵ ỡ Xij ỵ ỡ tữỡ ự ữợ ✸✿ ❇ê s✉♥❣ ✈➔♦ P Xij t➜t ❝↔ ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ P ♠➔ ❦❤æ♥❣ ♣❤↔✐ ❧➔ q✉② t➢❝ ✤ì♥✳ ❱➼ ❞ư ✸✳✺✳ G❂< Σ, ∆, S, P > ✈ỵ✐✿ Σ = {a, b, c}✱ ∆ = {S, A, B, C}✱ P = {A → B, A → C, C → a, A → c, B → b, S → AabA} • A ❈❤♦ ✈➠♥ ♣❤↕♠ A → B, A → C ✳ S → AabA ❧➔✿ ỵ ỡ ợ q t ỡ ❝→❝ q✉② t➢❝ s✐♥❤ s✐♥❤ r❛ tø ♣❤➨♣ t❤➳ ✣÷❛ ✈➔♦ P S → BabA | AabB | BabB | CabA | AabC | CabC | BabC | CabB • ❇ê s✉♥❣ ✈➔♦ P ❝→❝ q✉② t➢❝ s✐♥❤ tr♦♥❣ P ♠➔ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ q✉② t➢❝ ✤ì♥ t❛ ✤÷đ❝✿ P = {A → c, B → b, c → a, S → BabA | AabB | BabB | CabA | AabC | CabC | BabC | CabB} ❱➟② G ❂< Σ, ∆, S, P > ❧➔ ✈➠♥ ♣❤↕♠ ❝➛♥ t➻♠ ✈➔ L (G ) = L (G) ▼é✐ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ G❂< Σ, ∆, S, P > tỗ t ỳ tữỡ ữỡ ✭s❛✐ ❦❤→❝ tø ré♥❣✮ ❦❤æ♥❣ ❝â ε✲q✉② t➢❝✱ ❦❤æ♥❣ ❝â q t ỡ ổ õ ỵ tứ ự ♠✐♥❤✳ ✣à♥❤ ❧➼ ✸✳✶✳ ❱✐➺❝ ①→❝ ✤à♥❤ ✈➠♥ ♣❤↕♠ ♣❤✐ ỳ tữỡ ữỡ tỹ t ữợ s ữợ ỳ tứ ré♥❣✮ ✈➔ G ❦❤ỉ♥❣ ❝❤ù❛ ε✲q✉② t➢❝✳ G t÷ì♥❣ ✤÷ì♥❣ ợ G s ữỡ P P ữợ ỳ G tữỡ ữỡ ợ G G ổ ự q t ỡ ữợ ỳ ổ ự ỵ tứ G G tữỡ ữỡ ợ G G t➻♠✳ ▼é✐ ♥❣æ♥ ♥❣ú ♣❤✐ ♥❣ú ❝↔♥❤ ❦❤→❝ ré♥❣ ✤➲✉ ❝â t❤➸ ✤÷đ❝ s✐♥❤ r❛ ❜ð✐ ♠ët ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ❦❤æ♥❣ ❝â ε✲q✉② t➢❝✱ ❦❤æ♥❣ ❝â q✉② t➢❝ ỡ ổ õ ỵ tứ ự G < Σ, ∆, S, P > L (G) ✣à♥❤ ❧➼ ✸✳✷✳ ●å✐ ❂ ❧➔ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✈➔ ❧➔ ♥❣æ♥ ♥❣ú ❝õ❛ ♥â✳ ❑❤æ♥❣ ❣✐↔♠ t➼♥❤ tê♥❣ q✉→t t❛ ❣✐↔ sû ✈➠♥ ♣❤↕♠ ❦❤æ♥❣ ❝â ε✲q✉② t➢❝ ✈➔ ổ õ ỵ tứ ỹ ♥❣ú ❝↔♥❤ ♥❣ú G L\ {ε} = L (G ) tr♦♥❣ ✤â G G ❂< Σ, ∆ , S, P > s r ổ ổ õ ỵ tứ ổ ❝â ε✲q✉② t➢❝ ✈➔ ❦❤ỉ♥❣ ❝â q✉② t➢❝ ✤ì♥✳ ✣➦t • P ♥❤÷ s❛✉✿ ◆➳✉ tr♦♥❣ ∆ P S A → B ✱ ✈ỵ✐ A, B ∈ ∆ ❦❤✐ ✤â s tỗ t B ợ , (Σ ∪ ∆)∗ ✱ ω ∈ Σ ✭❞♦ αAβ ❦❤æ♥❣ ự ỵ tứ A AB tr P ❝→❝ q✉② t➢❝ s✐♥❤ S → αAβ ✤➲✉ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❝→❝ q✉② t➢❝ ✤ì♥ ♥❤÷♥❣ ❝❤ù❝ ♥➠♥❣ s✐♥❤ ♥❣ỉ♥ ỳ tữỡ ữỡ ợ q t s õ q✉② t➢❝ s✐♥❤ ❚❤❛② q✉② t➢❝ s✐♥❤ ✈➔ P ✳ t ỹ ữ tt q t➢❝ s✐♥❤ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ q✉② t➢❝ ✤ì♥ tr♦♥❣ P • ∆ = ∆✳ a → B✳ L (G ) = L − {ε}✳ ✸✳✸ ❉↕♥❣ ❝❤✉➞♥ ❈❤♦♠s❦② ✸✳✸✳✶✳ ❱➠♥ ♣❤↕♠ ❝❤✉➞♥ ❝õ❛ ❈❤♦♠s❦② ✣à♥❤ ♥❣❤➽❛ ✸✳✻✳ ❱➠♥ ♣❤↕♠ ♣❤✐ ỳ G ổ õ ỵ A BC ❝â ♣❤↕♠ ❝❤✉➞♥ ❝õ❛ ❈❤♦♠s❦② ♥➳✉ tr♦♥❣ P ✤➲✉ a ∈ Σ✳ q✉② t➢❝ s✐♥❤ tr♦♥❣ A, B, C ∈ ∆ ✈➔ ❝â ❞↕♥❣ G❂< Σ, ∆, S, P > ❣å✐ ❧➔ ✈➠♥ ❤✐➺✉ t❤ø❛ ✈➔ ♠å✐ ❞↕♥❣ A → a ợ ữỡ P P ①➨t✿ ✸✸ ❈→❝ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ð ❞↕♥❣ ❝❤✉➞♥ ❈❤♦♠s❦② ✤➲✉ ❧➔ ε✲q✉② t➢❝✳ ❈❤♦ ♥➯♥ ❝→❝ ♥❣æ♥ ♥❣ú ❞♦ ❝→❝ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ❦❤æ♥❣ ❝â ❝❤ó♥❣ s✐♥❤ r❛ ✤➲✉ ❦❤ỉ♥❣ ❝❤ù❛ ①➙✉ ré♥❣✳ ❱➼ ❞ư ✸✳✻✳ ❱➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ G❂< Σ, ∆, S, P > ✈ỵ✐✿ Σ = {a, b, c}✱ ∆ = {S, A, B, C, F }✱ P = {A → AB, A → BF, F → BS, A → b, B → c, C → a} ❧➔ ✈➠♥ ♣❤↕♠ ❞↕♥❣ ❝❤✉➞♥ ❈❤♦♠s❦②✳ ✸✳✸✳✷✳ ✣÷❛ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✈➲ ❞↕♥❣ ❝❤✉➞♥ ❈❤♦♠s❦② ✣è✐ ✈ỵ✐ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ tũ ỵ G< , , S, P ổ tỗ t ♠ët ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ð ❞↕♥❣ ❝❤✉➞♥ ❈❤♦♠s❦② G ❂< Σ, ∆ , S, P > t÷ì♥❣ ✤÷ì♥❣ ợ G ự > ữợ ọ q t ỡ t ữợ tt t♦→♥✳ G ❂< Σ, ∆ , S, P > tø G ❝❤➾ ❝❤ù❛ A ←− B1 B2 Bn ✈ỵ✐ Bi , a ữợ ỹ ✈➠♥ ♣❤↕♠ A→a t➢❝ s✐♥❤ ❝â ❞↕♥❣✿ • ✣➦t ∆ = ∆✳ • ◆➳✉ A→X ✈➔ ❧➔ q✉② t➢❝ s✐♥❤ tr♦♥❣ P ✱ |X| ≥ ✈➔ X q✉② ❝❤ù❛ ỵ ỡ t s ✈➔♦ ✣÷❛ ✈➔♦ P ∆ {¯ a | a ∈ } tr õ X AX ỵ ❤✐➺✉ ♣❤õ ✤à♥❤ q✉② t➢❝ s✐♥❤ t❤➔♥❤ tø X ❜➡♥❣ t ỵ ỡ ợ ỵ tữỡ ự s P t➟♣ q✉② t➢❝ s✐♥❤ P ❝→❝ q✉② t➢❝ s✐♥❤ A → B1 B2 Bn ✈ỵ✐ Bi ∈ ∆ ❇ê s✉♥❣ ✈➔♦ {¯ a → a | a ∈ Σ} tr♦♥❣ P ❝â ❞↕♥❣ A→a G ❂< Σ, ∆ , S, P > tø G ❝❤➾ A → a ✈ỵ✐ A, B, C , a ữợ ✸✿ ❳➙② ❞ü♥❣ ✈➠♥ ♣❤↕♠ t➢❝ s✐♥❤ ❝â ❞↕♥❣ A BC rữợ t t t ✈➔ ❝❤ù❛ q✉② ∆ =∆ A → B1 B2 Bn ✈ỵ✐ (n ≥ 2) ❧➔ q✉② t➢❝ s✐♥❤ tr♦♥❣ P t❤➻✿ ❈❤÷ì♥❣ ✸✳ ✕ ❱❿◆ P❍❸▼ P❍■ ◆●Ú ❈❷◆❍ ∆ s ỵ ợ K1 , K2 , Kn2 ộ s ỵ ợ P q t➢❝ s✐♥❤✿ A → B1 K1 , K1 → B2 K2 , , Kn−2 → Bn−1 Bn ✣÷❛ ✈➔♦ ❇ê s✉♥❣ ✈➔♦ P ❝→❝ q✉② t➢❝ s✐♥❤ ❝õ❛ A → a, A → BC ✈ỵ✐ P ❝â ❞↕♥❣✿ A, B, C ∈ ∆ , a ∈ Σ ❱➟② t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ð ❞↕♥❣ s tữỡ ữỡ ợ G G ❈❤♦ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ G❂< Σ, ∆, S, P > tr♦♥❣ ✤â✿ Σ = {a, b, c, d}✱ ∆ = {S, A, B} P = {S → Sa, S → ABb, A → ABBa, B → b, A → d, S → c} ◗✉→ tr➻♥❤ ❜✐➳♥ ✤ê✐ G t s ữ s ữợ G ❂< Σ, ∆ , S, P > tr♦♥❣ ✤â✿ ∆ = S, A, B, a ¯, ¯b ✱ P = {S → S¯ a, a ¯ → a, S → AB¯b, A → ABB¯ a, ¯b → b, B b, A d, S c} ữợ ✷✿ G ❂ < Σ, ∆ , S, P > ✈ỵ✐✿ ∆ = S, A, B, a ¯, ¯b, K1 , S1 , S2 P = {S → S¯ a, a ¯ → a, S → AK1 , K1 → B¯b, ¯b → b, A → AS1 , S1 → BS2 , S2 → B¯ a, B → b, A → d, S → c} ✣➳♥ ✤➙② t❛ ✤÷đ❝ ✈➠♥ s G tữỡ ữỡ ợ G ▲❯❾◆ ❑❤♦→ ❧✉➟♥ t➻♠ ❤✐➸✉ ✈➲ ❱➠♥ ♣❤↕♠ ✈➔ ♥❣æ♥ ♥❣ú ♣❤✐ ♥❣ú ❝↔♥❤✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ✤↕t ✤÷đ❝ ❧➔✿ ✶✳ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ♥❣ỉ♥ ♥❣ú ❤➻♥❤ t❤ù❝ ❝ò♥❣ ✈ỵ✐ ❝→❝ ♣❤➨♣ t♦→♥ tr➯♥ tø ✈➔ tr➯♥ ♥❣æ♥ ♥❣ú✳ ✷✳ ❚r➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ ✈➠♥ ♣❤↕♠✱ q✉→ tr➻♥❤ s✐♥❤ r❛ ♥❣æ♥ ♥❣ú ❝õ❛ ✈➠♥ ♣❤↕♠✱ t➼♥❤ ❝❤➜t ✤â♥❣ ❝õ❛ ❧ỵ♣ ♥❣ỉ♥ ♥❣ú ♣❤✐ ♥❣ú ❝↔♥❤ ✈ỵ✐ ❝→❝ ♣❤➨♣ ❤đ♣✱ ❣✐❛♦✱ ❣❤➨♣✱ ❧➦♣✱ ❧➦♣ ❝➢t✱ ❝❤✐❛ tr→✐ ✈➔ ❝❤✐❛ ♣❤↔✐✳ ✸✳ ❚r➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ❝õ❛ ✈➠♥ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✈➔ ♠ët số ữỡ ữủ ú ỗ tớ ữ ♣❤↕♠ ♣❤✐ ♥❣ú ❝↔♥❤ ✈➲ ❞↕♥❣ ❝❤✉➞♥ ❈❤♦♠s❦②✳ ❚✉② ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ✈➔ ❦❤↔ ♥➠♥❣ ❝â ❤↕♥ ♥➯♥ ❝→❝ ✈➜♥ ✤➲ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝á♥ ❝❤÷❛ ✤÷đ❝ tr➻♥❤ ❜➔② s➙✉ s➢❝ ✈➔ ✈➝♥ ❝á♥ ỳ t sõt ữủ sỹ õ ỵ ①➙② ❞ü♥❣ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✸✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ✣à♥❤ ✭✷✵✶✷✮✱ ●✐→♦ tr➻♥❤✿ ❖t♦♠❛t ✈➔ ♥❣æ♥ ♥❣ú ❤➻♥❤ t❤ù❝ ✱ ◆❳❇ ✣↕✐ ❤å❝ ◆æ♥❣ ◆❣❤✐➺♣✳ ❬✷❪ ❏♦❤♥ ❊✳ ❍♦♣❝r♦❢t ✲ ❘❛❥❡❡✈ ▼♦t✇❛♥✐ ✲ ❏❡❢❢r❡② ❉✳❯❧❧♠❛♥ ✭✷✵✵✶✮✱ tr♦❞✉❝t✐♦♥ t♦ ❆✉t♦♠❛t❛ ❚❤❡♦r② ▲❛♥❣✉❛❣❡s✱ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ■♥✲ ■♥❝✱ P ỗ tự tr ỵ t❤✉②➳t ➷tæ♠→t ✈➔ ♥❣æ♥ ♥❣ú ❤➻♥❤ ✱ ◆❳❇ ✣❍ ◗✉è❝ ●✐❛ ❚♣✳ ❍❈▼✳ ✸✻ ... HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* BÙI PHƯƠNG HIỀN VĂN PHẠM VÀ NGÔN NGỮ PHI NGỮ CẢNH KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Toán ứng dụng Người hướng dẫn khoa học TS KIỀU VĂN HƯNG