HNU 编译系统 作业2（3）

题目3（续）

（3）（a|b）*a（a|b）（a|b）（a|b）
e -> (a|b)*a(a|b)(a|b)(a|b)

利用子集构造算法将NFA转为DFA：
初始状态：{s0}:q0{q0}→ε{s0,s1,s2,s4,s7,s8}:q0{q0}→a{s3,s9}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s13}:q1{q0}→b{s5}→ε{s1,s2,s4,s5,s6,s7,s8}:q2{q1}→a{s3,s9,s12}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s12,s13,s15,s16,s18}:q3{q1}→b{s5,s14}→ε{s1,s2,s4,s5,s6,s7,s8,s14,s15,s16,s18}:q4{q2}→a{s3,s9}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s13}:q1{q2}→b{s5}→ε{s1,s2,s4,s5,s6,s7,s8}:q2{q3}→a{s3,s9,s12,s17}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s12,s13,s15,s16,s17,s18,s20,s21,s23}:q5{q3}→b{s5,s14,s19}→ε{s1,s2,s4,s5,s6,s7,s8,s14,s15,s16,s18,s19,s20,s21,s23}:q6{q4}→a{s3,s9,s17}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s13,s17,s20,s21,s23}:q7{q4}→b{s5,s19}→ε{s1,s2,s4,s5,s6,s7,s8,s19,s20,s21,s23}:q8{q5}→a{s3,s9,s12,s17,s22}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s12,s13,s15,s16,s17,s18,s20,s21,s22,s23,s25}:q9{q5}→b{s5,s14,s19,s24}→ε{s1,s2,s4,s5,s6,s7,s8,s14,s15,s16,s18,s19,s20,s21,s23,s24,s25}:q10{q6}→a{s3,s9,s17,s22}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s13,s17,s20,s21,s22,s23,s25}:q11{q6}→b{s5,s19,s24}→ε{s1,s2,s4,s5,s6,s7,s8,s19,s20,s21,s23,s24,s25}:q12{q7}→a{s3,s9,s12,s22}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s12,s13,s15,s16,s18,s22,s25}:q13{q7}→b{s5,s14,s24}→ε{s1,s2,s4,s5,s6,s7,s8,s14,s15,s16,s18,s24,s25}:q14{q8}→a{s3,s9,s22}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s13,s22,s25}:q15{q8}→b{s5,s24}→ε{s1,s2,s4,s5,s6,s7,s8,s24,s25}:q16{q9}→a{s3,s9,s12,s17,s22}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s12,s13,s15,s16,s17,s18,s20,s21,s22,s23,s25}:q9{q9}→b{s5,s14,s19,s24}→ε{s1,s2,s4,s5,s6,s7,s8,s14,s15,s16,s18,s19,s20,s21,s23,s24,s25}:q10{q10}→a{s3,s9,s17,s22}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s13,s17,s20,s21,s22,s23,s25}:q11{q10}→b{s5,s19,s24}→ε{s1,s2,s4,s5,s6,s7,s8,s19,s20,s21,s23,s24,s25}:q12{q11}→a{s3,s9,s12,s22}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s12,s13,s15,s16,s18,s22,s25}:q13{q11}→b{s5,s14,s24}→ε{s1,s2,s4,s5,s6,s7,s8,s14,s15,s16,s18,s24,s25}:q14{q12}→a{s3,s9,s22}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s13,s22,s25}:q15{q12}→b{s5,s24}→ε{s1,s2,s4,s5,s6,s7,s8,s24,s25}:q16{q13}→a{s3,s9,s12,s17}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s12,s13,s15,s16,s17,s18,s20,s21,s23}:q5{q13}→b{s5,s14,s19}→ε{s1,s2,s4,s5,s6,s7,s8,s14,s15,s16,s18,s19,s20,s21,s23}:q6{q14}→a{s3,s9,s17}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s13,s17,s20,s21,s23}:q7{q14}→b{s5,s19}→ε{s1,s2,s4,s5,s6,s7,s8,s19,s20,s21,s23}:q8{q15}→a{s3,s9,s12}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s12,s13,s15,s16,s18}:q3{q15}→b{s5,s14}→ε{s1,s2,s4,s5,s6,s7,s8,s14,s15,s16,s18}:q4{q16}→a{s3,s9}→ε{s1,s2,s3,s4,s6,s7,s8,s9,s10,s11,s13}:q1{q16}→b{s5}→ε{s1,s2,s4,s5,s6,s7,s8}:q2 \begin{aligned} &初始状态：\{s_0\}:q_0\\ &\{q_0\}\xrightarrow{ε}\{s_0, s_1, s_2, s_4, s_7, s_8\}:q_0\\ &\{q_0\}\xrightarrow{a}\{s_3, s_9\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{13}\}:q_1\\ &\{q_0\}\xrightarrow{b}\{s_5\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8\}:q_2\\ &\{q_1\}\xrightarrow{a}\{s_3, s_9, s_{12}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{12}, s_{13}, s_{15}, s_{16}, s_{18}\}:q_3\\ &\{q_1\}\xrightarrow{b}\{s_5, s_{14}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{14}, s_{15}, s_{16}, s_{18}\}:q_4\\ &\{q_2\}\xrightarrow{a}\{s_3, s_9\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{13}\}:q_1\\ &\{q_2\}\xrightarrow{b}\{s_5\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8\}:q_2\\ &\{q_3\}\xrightarrow{a}\{s_3, s_9, s_{12}, s_{17}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{12}, s_{13}, s_{15}, s_{16}, s_{17}, s_{18}, s_{20}, s_{21}, s_{23}\}:q_5\\ &\{q_3\}\xrightarrow{b}\{s_5, s_{14}, s_{19}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{14}, s_{15}, s_{16}, s_{18}, s_{19}, s_{20}, s_{21}, s_{23}\}:q_6\\ &\{q_4\}\xrightarrow{a}\{s_3, s_9, s_{17}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{13}, s_{17}, s_{20}, s_{21}, s_{23}\}:q_7\\ &\{q_4\}\xrightarrow{b}\{s_5, s_{19}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{19}, s_{20}, s_{21}, s_{23}\}:q_8\\ &\{q_5\}\xrightarrow{a}\{s_3, s_9, s_{12}, s_{17}, s_{22}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{12}, s_{13}, s_{15}, s_{16}, s_{17}, s_{18}, s_{20}, s_{21}, s_{22}, s_{23}, s_{25}\}:q_9\\ &\{q_5\}\xrightarrow{b}\{s_5, s_{14}, s_{19}, s_{24}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{14}, s_{15}, s_{16}, s_{18}, s_{19}, s_{20}, s_{21}, s_{23}, s_{24}, s_{25}\}:q_{10}\\ &\{q_6\}\xrightarrow{a}\{s_3, s_9, s_{17}, s_{22}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{13}, s_{17}, s_{20}, s_{21}, s_{22}, s_{23}, s_{25}\}:q_{11}\\ &\{q_6\}\xrightarrow{b}\{s_5, s_{19}, s_{24}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{19}, s_{20}, s_{21}, s_{23}, s_{24}, s_{25}\}:q_{12}\\ &\{q_7\}\xrightarrow{a}\{s_3, s_9, s_{12}, s_{22}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{12}, s_{13}, s_{15}, s_{16}, s_{18}, s_{22}, s_{25}\}:q_{13}\\ &\{q_7\}\xrightarrow{b}\{s_5, s_{14}, s_{24}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{14}, s_{15}, s_{16}, s_{18}, s_{24}, s_{25}\}: q_{14}\\ &\{q_8\}\xrightarrow{a}\{s_3, s_9, s_{22}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{13}, s_{22}, s_{25}\}:q_{15}\\ &\{q_8\}\xrightarrow{b}\{s_5, s_{24}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{24}, s_{25}\}:q_{16}\\ &\{q_9\}\xrightarrow{a}\{s_3, s_9, s_{12}, s_{17}, s_{22}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{12}, s_{13}, s_{15}, s_{16}, s_{17}, s_{18}, s_{20}, s_{21}, s_{22}, s_{23}, s_{25}\}:q_9\\ &\{q_9\}\xrightarrow{b}\{s_5, s_{14}, s_{19}, s_{24}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{14}, s_{15}, s_{16}, s_{18}, s_{19}, s_{20}, s_{21}, s_{23}, s_{24}, s_{25}\}:q_{10}\\ &\{q_{10}\}\xrightarrow{a}\{s_3, s_9, s_{17}, s_{22}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{13}, s_{17}, s_{20}, s_{21}, s_{22}, s_{23}, s_{25}\}:q_{11}\\ &\{q_{10}\}\xrightarrow{b}\{s_5, s_{19}, s_{24}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{19}, s_{20}, s_{21}, s_{23}, s_{24}, s_{25}\}:q_{12}\\ &\{q_{11}\}\xrightarrow{a}\{s_3, s_9, s_{12}, s_{22}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{12}, s_{13}, s_{15}, s_{16}, s_{18}, s_{22}, s_{25}\}:q_{13}\\ &\{q_{11}\}\xrightarrow{b}\{s_5, s_{14}, s_{24}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{14}, s_{15}, s_{16}, s_{18}, s_{24}, s_{25}\}: q_{14}\\ &\{q_{12}\}\xrightarrow{a}\{s_3, s_9, s_{22}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{13}, s_{22}, s_{25}\}:q_{15}\\ &\{q_{12}\}\xrightarrow{b}\{s_5, s_{24}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{24}, s_{25}\}:q_{16}\\ &\{q_{13}\}\xrightarrow{a}\{s_3, s_9, s_{12}, s_{17}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{12}, s_{13}, s_{15}, s_{16}, s_{17}, s_{18}, s_{20}, s_{21}, s_{23}\}:q_5\\ &\{q_{13}\}\xrightarrow{b}\{s_5, s_{14}, s_{19}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{14}, s_{15}, s_{16}, s_{18}, s_{19}, s_{20}, s_{21}, s_{23}\}:q_6\\ &\{q_{14}\}\xrightarrow{a}\{s_3, s_9, s_{17}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{13}, s_{17}, s_{20}, s_{21}, s_{23}\}:q_7\\ &\{q_{14}\}\xrightarrow{b}\{s_5, s_{19}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{19}, s_{20}, s_{21}, s_{23}\}:q_8\\ &\{q_{15}\}\xrightarrow{a}\{s_3, s_9, s_{12}\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{12}, s_{13}, s_{15}, s_{16}, s_{18}\}:q_3\\ &\{q_{15}\}\xrightarrow{b}\{s_5, s_{14}\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8, s_{14}, s_{15}, s_{16}, s_{18}\}:q_4\\ &\{q_{16}\}\xrightarrow{a}\{s_3, s_9\}\xrightarrow{ε}\{s_1, s_2, s_3, s_4, s_6, s_7, s_8, s_9, s_{10}, s_{11}, s_{13}\}:q_1\\ &\{q_{16}\}\xrightarrow{b}\{s_5\}\xrightarrow{ε}\{s_1, s_2, s_4, s_5, s_6, s_7, s_8\}:q_2 \end{aligned} ​初始状态：{s0​}:q0​{q0​}ε​{s0​,s1​,s2​,s4​,s7​,s8​}:q0​{q0​}a​{s3​,s9​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s13​}:q1​{q0​}b​{s5​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​}:q2​{q1​}a​{s3​,s9​,s12​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s12​,s13​,s15​,s16​,s18​}:q3​{q1​}b​{s5​,s14​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s14​,s15​,s16​,s18​}:q4​{q2​}a​{s3​,s9​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s13​}:q1​{q2​}b​{s5​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​}:q2​{q3​}a​{s3​,s9​,s12​,s17​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s12​,s13​,s15​,s16​,s17​,s18​,s20​,s21​,s23​}:q5​{q3​}b​{s5​,s14​,s19​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s14​,s15​,s16​,s18​,s19​,s20​,s21​,s23​}:q6​{q4​}a​{s3​,s9​,s17​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s13​,s17​,s20​,s21​,s23​}:q7​{q4​}b​{s5​,s19​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s19​,s20​,s21​,s23​}:q8​{q5​}a​{s3​,s9​,s12​,s17​,s22​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s12​,s13​,s15​,s16​,s17​,s18​,s20​,s21​,s22​,s23​,s25​}:q9​{q5​}b​{s5​,s14​,s19​,s24​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s14​,s15​,s16​,s18​,s19​,s20​,s21​,s23​,s24​,s25​}:q10​{q6​}a​{s3​,s9​,s17​,s22​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s13​,s17​,s20​,s21​,s22​,s23​,s25​}:q11​{q6​}b​{s5​,s19​,s24​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s19​,s20​,s21​,s23​,s24​,s25​}:q12​{q7​}a​{s3​,s9​,s12​,s22​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s12​,s13​,s15​,s16​,s18​,s22​,s25​}:q13​{q7​}b​{s5​,s14​,s24​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s14​,s15​,s16​,s18​,s24​,s25​}:q14​{q8​}a​{s3​,s9​,s22​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s13​,s22​,s25​}:q15​{q8​}b​{s5​,s24​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s24​,s25​}:q16​{q9​}a​{s3​,s9​,s12​,s17​,s22​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s12​,s13​,s15​,s16​,s17​,s18​,s20​,s21​,s22​,s23​,s25​}:q9​{q9​}b​{s5​,s14​,s19​,s24​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s14​,s15​,s16​,s18​,s19​,s20​,s21​,s23​,s24​,s25​}:q10​{q10​}a​{s3​,s9​,s17​,s22​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s13​,s17​,s20​,s21​,s22​,s23​,s25​}:q11​{q10​}b​{s5​,s19​,s24​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s19​,s20​,s21​,s23​,s24​,s25​}:q12​{q11​}a​{s3​,s9​,s12​,s22​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s12​,s13​,s15​,s16​,s18​,s22​,s25​}:q13​{q11​}b​{s5​,s14​,s24​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s14​,s15​,s16​,s18​,s24​,s25​}:q14​{q12​}a​{s3​,s9​,s22​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s13​,s22​,s25​}:q15​{q12​}b​{s5​,s24​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s24​,s25​}:q16​{q13​}a​{s3​,s9​,s12​,s17​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s12​,s13​,s15​,s16​,s17​,s18​,s20​,s21​,s23​}:q5​{q13​}b​{s5​,s14​,s19​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s14​,s15​,s16​,s18​,s19​,s20​,s21​,s23​}:q6​{q14​}a​{s3​,s9​,s17​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s13​,s17​,s20​,s21​,s23​}:q7​{q14​}b​{s5​,s19​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s19​,s20​,s21​,s23​}:q8​{q15​}a​{s3​,s9​,s12​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s12​,s13​,s15​,s16​,s18​}:q3​{q15​}b​{s5​,s14​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​,s14​,s15​,s16​,s18​}:q4​{q16​}a​{s3​,s9​}ε​{s1​,s2​,s3​,s4​,s6​,s7​,s8​,s9​,s10​,s11​,s13​}:q1​{q16​}b​{s5​}ε​{s1​,s2​,s4​,s5​,s6​,s7​,s8​}:q2​​

将DFA简化到最小：

字符a可以分裂DFA，对于N中状态：δ(q0​,a)=q1∈Nδ(q1​,a)=q3∈Nδ(q2​,a)=q1∈Nδ(q3​,a)=q5∈Nδ(q4,a)=q7∈Nδ(q5​,a)=q9∉Nδ(q6​,a)=q11∉Nδ(q7,a)=q13∉Nδ(q8,a)=q15∉N对于A中状态：δ(q9​,a)=q9∈Aδ(q10​,a)=q11∈Aδ(q11​,a)=q13∈Aδ(q12,a)=q15∈Aδ(q13​,a)=q5∉Aδ(q14​,a)=q7∉Aδ(q15​,a)=q3∉Aδ(q16,a)=q1∉A因此字符a可将原DFA分裂为{q0,q1,q2,q3,q4},{q5,q6,q7,q8},{q9,q10,q11,q12},{q13,q14,q15,q16}字符b可以继续分裂DFA：δ(q0​,b)=q2∈{q0,q1,q2,q3,q4}δ(q1​,b)=q4∈{q0,q1,q2,q3,q4}δ(q2​,b)=q2∈{q0,q1,q2,q3,q4}δ(q3​,b)=q6∈{q5,q6,q7,q8}δ(q4,b)=q8∈{q5,q6,q7,q8}δ(q5​,b)=q10∈{q9,q10,q11,q12}δ(q6​,b)=q12∈{q9,q10,q11,q12}δ(q7​,b)=q14∈{q13,q14,q15,q16}δ(q8​,b)=q16∈{q13,q14,q15,q16}δ(q9​,b)=q10∈{q9,q10,q11,q12}δ(q10​,b)=q12∈{q9,q10,q11,q12}δ(q11​,b)=q14∈{q13,q14,q15,q16}δ(q12,b)=q16∈{q13,q14,q15,q16}δ(q13​,b)=q6∈{q5,q6,q7,q8}δ(q14​,b)=q8∈{q5,q6,q7,q8}δ(q15​,b)=q4∈{q0,q1,q2,q3,q4}δ(q16,b)=q2∈{q0,q1,q2,q3,q4}因此字符b可将原DFA继续分裂为{q0,q1,q2},{q3,4},{q5,q6},{q7,q8},{q9,q10},{q11,q12},{q13,q14},{q15,q16}字符b可以继续分裂DFA：δ(q0​,b)=q2∈{q0,q1,q2}δ(q1​,b)=q4∈{q3,q4}δ(q2​,b)=q2∈{q0,q1,q2}δ(q3​,b)=q6∈{q5,q6}δ(q4,b)=q8∈{q7,q8}δ(q5​,b)=q10∈{q9,q10}δ(q6​,b)=q12∈{q11,q12}δ(q7​,b)=q14∈{q13,q14}δ(q8​,b)=q16∈{q15,q16}δ(q9​,b)=q10∈{q9,q10}δ(q10​,b)=q12∈{q11,q12}δ(q11​,b)=q14∈{q13,q14}δ(q12,b)=q16∈{q15,q16}δ(q13​,b)=q6∈{q5,q6}δ(q14​,b)=q8∈{q7,q8}δ(q15​,b)=q4∈{q3,q4}δ(q16,b)=q2∈{q0,q1,q2}因此字符b可将原DFA继续分裂为{q0,q2},{q1},{q3},{q4},{q5},{q6},{q7},{q8},{q9},{q10},{q11},{q12},{q13},{q14},{q15},{q16}此后字符a和b都不能继续分裂DFA。因此最简的DFA： \begin{aligned} &字符a可以分裂DFA，对于N中状态：\\ &δ(q_0​,a)=q_1 \in N \\ &δ(q_1​,a)=q_3 \in N \\ &δ(q_2​,a)=q_1 \in N \\ &δ(q_3​,a)=q_5 \in N \\ &δ(q_4,a)=q_7 \in N \\ &δ(q_5​,a)=q_9 \notin N \\ &δ(q_6​,a)=q_{11} \notin N \\ &δ(q_7,a)=q_{13} \notin N \\ &δ(q_8,a)=q_{15} \notin N \\ &对于A中状态：\\ &δ(q_{9}​,a)=q_{9} \in A \\ &δ(q_{10}​,a)=q_{11} \in A \\ &δ(q_{11}​,a)=q_{13} \in A \\ &δ(q_{12},a)=q_{15} \in A \\ &δ(q_{13}​,a)=q_{5} \notin A \\ &δ(q_{14}​,a)=q_{7} \notin A \\ &δ(q_{15}​,a)=q_{3} \notin A \\ &δ(q_{16},a)=q_{1} \notin A \\ &因此字符a可将原DFA分裂为\{q_0, q_1, q_2, q_3, q_4\}, \{q_5, q_6, q_7, q_8\}, \{q_9, q_{10}, q_{11}, q_{12}\}, \{q_{13}, q_{14}, q_{15}, q_{16}\}\\ &字符b可以继续分裂DFA：\\ &δ(q_0​,b)=q_2 \in \{q_0, q_1, q_2, q_3, q_4\} \\ &δ(q_1​,b)=q_4 \in \{q_0, q_1, q_2, q_3, q_4\} \\ &δ(q_2​,b)=q_2 \in \{q_0, q_1, q_2, q_3, q_4\} \\ &δ(q_3​,b)=q_6 \in \{q_5, q_6, q_7, q_8\} \\ &δ(q_4,b)=q_8 \in \{q_5, q_6, q_7, q_8\} \\ &δ(q_5​,b)=q_{10} \in \{q_9, q_{10}, q_{11}, q_{12}\} \\ &δ(q_6​,b)=q_{12} \in \{q_9, q_{10}, q_{11}, q_{12}\} \\ &δ(q_7​,b)=q_{14} \in \{q_{13}, q_{14}, q_{15}, q_{16}\} \\ &δ(q_8​,b)=q_{16} \in \{q_{13}, q_{14}, q_{15}, q_{16}\} \\ &δ(q_{9}​,b)=q_{10} \in \{q_9, q_{10}, q_{11}, q_{12}\} \\ &δ(q_{10}​,b)=q_{12} \in \{q_9, q_{10}, q_{11}, q_{12}\} \\ &δ(q_{11}​,b)=q_{14} \in \{q_{13}, q_{14}, q_{15}, q_{16}\} \\ &δ(q_{12},b)=q_{16} \in \{q_{13}, q_{14}, q_{15}, q_{16}\} \\ &δ(q_{13}​,b)=q_6 \in \{q_5, q_6, q_7, q_8\} \\ &δ(q_{14}​,b)=q_8 \in \{q_5, q_6, q_7, q_8\} \\ &δ(q_{15}​,b)=q_4 \in \{q_0, q_1, q_2, q_3, q_4\} \\ &δ(q_{16},b)=q_2 \in \{q_0, q_1, q_2, q_3, q_4\} \\ &因此字符b可将原DFA继续分裂为\{q_0, q_1, q_2\}, \{q_3, _4\}, \{q_5, q_6\}, \{q_7, q_8\}, \{q_9, q_{10}\}, \{q_{11}, q_{12}\}, \{q_{13}, q_{14}\}, \{q_{15}, q_{16}\}\\ &字符b可以继续分裂DFA：\\ &δ(q_0​,b)=q_2 \in \{q_0, q_1, q_2\} \\ &δ(q_1​,b)=q_4 \in \{q_3, q_4\} \\ &δ(q_2​,b)=q_2 \in \{q_0, q_1, q_2\} \\ &δ(q_3​,b)=q_6 \in \{q_5, q_6\} \\ &δ(q_4,b)=q_8 \in \{q_7, q_8\} \\ &δ(q_5​,b)=q_{10} \in \{q_9, q_{10}\} \\ &δ(q_6​,b)=q_{12} \in \{q_{11}, q_{12}\} \\ &δ(q_7​,b)=q_{14} \in \{q_{13}, q_{14}\} \\ &δ(q_8​,b)=q_{16} \in \{q_{15}, q_{16}\} \\ &δ(q_{9}​,b)=q_{10} \in \{q_9, q_{10}\} \\ &δ(q_{10}​,b)=q_{12} \in \{q_{11}, q_{12}\} \\ &δ(q_{11}​,b)=q_{14} \in \{q_{13}, q_{14}\} \\ &δ(q_{12},b)=q_{16} \in \{q_{15}, q_{16}\} \\ &δ(q_{13}​,b)=q_6 \in \{q_5, q_6\} \\ &δ(q_{14}​,b)=q_8 \in \{q_7, q_8\} \\ &δ(q_{15}​,b)=q_4 \in \{q_3, q_4\} \\ &δ(q_{16},b)=q_2 \in \{q_0, q_1, q_2\} \\ &因此字符b可将原DFA继续分裂为\{q_0, q_2\}, \{q_1\}, \{q_3\}, \{q_4\}, \{q_5\}, \{q_6\}, \{q_7\}, \{q_8\}, \{q_9\}, \{q_{10}\}, \{q_{11}\}, \{q_{12}\}, \{q_{13}\}, \{q_{14}\}, \{q_{15}\}, \{q_{16}\}\\ &此后字符a和b都不能继续分裂DFA。因此最简的DFA： \end{aligned} ​字符a可以分裂DFA，对于N中状态：δ(q0​​,a)=q1​∈Nδ(q1​​,a)=q3​∈Nδ(q2​​,a)=q1​∈Nδ(q3​​,a)=q5​∈Nδ(q4​,a)=q7​∈Nδ(q5​​,a)=q9​∈/Nδ(q6​​,a)=q11​∈/Nδ(q7​,a)=q13​∈/Nδ(q8​,a)=q15​∈/N对于A中状态：δ(q9​​,a)=q9​∈Aδ(q10​​,a)=q11​∈Aδ(q11​​,a)=q13​∈Aδ(q12​,a)=q15​∈Aδ(q13​​,a)=q5​∈/Aδ(q14​​,a)=q7​∈/Aδ(q15​​,a)=q3​∈/Aδ(q16​,a)=q1​∈/A因此字符a可将原DFA分裂为{q0​,q1​,q2​,q3​,q4​},{q5​,q6​,q7​,q8​},{q9​,q10​,q11​,q12​},{q13​,q14​,q15​,q16​}字符b可以继续分裂DFA：δ(q0​​,b)=q2​∈{q0​,q1​,q2​,q3​,q4​}δ(q1​​,b)=q4​∈{q0​,q1​,q2​,q3​,q4​}δ(q2​​,b)=q2​∈{q0​,q1​,q2​,q3​,q4​}δ(q3​​,b)=q6​∈{q5​,q6​,q7​,q8​}δ(q4​,b)=q8​∈{q5​,q6​,q7​,q8​}δ(q5​​,b)=q10​∈{q9​,q10​,q11​,q12​}δ(q6​​,b)=q12​∈{q9​,q10​,q11​,q12​}δ(q7​​,b)=q14​∈{q13​,q14​,q15​,q16​}δ(q8​​,b)=q16​∈{q13​,q14​,q15​,q16​}δ(q9​​,b)=q10​∈{q9​,q10​,q11​,q12​}δ(q10​​,b)=q12​∈{q9​,q10​,q11​,q12​}δ(q11​​,b)=q14​∈{q13​,q14​,q15​,q16​}δ(q12​,b)=q16​∈{q13​,q14​,q15​,q16​}δ(q13​​,b)=q6​∈{q5​,q6​,q7​,q8​}δ(q14​​,b)=q8​∈{q5​,q6​,q7​,q8​}δ(q15​​,b)=q4​∈{q0​,q1​,q2​,q3​,q4​}δ(q16​,b)=q2​∈{q0​,q1​,q2​,q3​,q4​}因此字符b可将原DFA继续分裂为{q0​,q1​,q2​},{q3​,4​},{q5​,q6​},{q7​,q8​},{q9​,q10​},{q11​,q12​},{q13​,q14​},{q15​,q16​}字符b可以继续分裂DFA：δ(q0​​,b)=q2​∈{q0​,q1​,q2​}δ(q1​​,b)=q4​∈{q3​,q4​}δ(q2​​,b)=q2​∈{q0​,q1​,q2​}δ(q3​​,b)=q6​∈{q5​,q6​}δ(q4​,b)=q8​∈{q7​,q8​}δ(q5​​,b)=q10​∈{q9​,q10​}δ(q6​​,b)=q12​∈{q11​,q12​}δ(q7​​,b)=q14​∈{q13​,q14​}δ(q8​​,b)=q16​∈{q15​,q16​}δ(q9​​,b)=q10​∈{q9​,q10​}δ(q10​​,b)=q12​∈{q11​,q12​}δ(q11​​,b)=q14​∈{q13​,q14​}δ(q12​,b)=q16​∈{q15​,q16​}δ(q13​​,b)=q6​∈{q5​,q6​}δ(q14​​,b)=q8​∈{q7​,q8​}δ(q15​​,b)=q4​∈{q3​,q4​}δ(q16​,b)=q2​∈{q0​,q1​,q2​}因此字符b可将原DFA继续分裂为{q0​,q2​},{q1​},{q3​},{q4​},{q5​},{q6​},{q7​},{q8​},{q9​},{q10​},{q11​},{q12​},{q13​},{q14​},{q15​},{q16​}此后字符a和b都不能继续分裂DFA。因此最简的DFA：​

题目4

写出下列语言的正则定义。
（1）仅由a和b这2个字符组成的字符串，其中a后面必跟b。
(b|ab)*

（2）仅由a和b这2个字符组成的字符串，其中a和b出现的次数都为偶数次。
(aa|bb|(ab|ba)(aa|bb)*(ab|ba))*

• 每次匹配要么是 aa/bb（不改变奇偶性）
• 要么是成对的 (ab|ba)...(ab|ba)（净变化为偶数次），中间...处可插入任意偶数长度片段aa/bb
• 最外层的 *允许重复匹配这些模式
• 最终保证整个字符串中 a和 b都出现偶数次

（3）仅由a和b这2个字符组成的字符串，其中不含子串abb。
b*(aa*b)*a*

• b*​​：匹配任意数量的b（包括0个）
• (aa*b)*​​：匹配0个或多个"a序列+b"的块
  • aa*b= 至少一个a后跟一个b，例如：ab, aab, aaab等
• a*​​：匹配任意数量的a（包括0个）