29 |
33 |
30 \begin{center} |
34 \begin{center} |
31 $r ::= \varnothing \;|\; \epsilon \;|\; c \;|\; r_1 + r_2 \;|\; r_1 \cdot r_2 \;|\; r^*$ |
35 $r ::= \varnothing \;|\; \epsilon \;|\; c \;|\; r_1 + r_2 \;|\; r_1 \cdot r_2 \;|\; r^*$ |
32 \end{center} |
36 \end{center} |
33 |
37 |
34 \item Given a deterministic finite automata $A(Q, q_0, F, \delta)$, |
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35 define which language is recognised by this automaton. |
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36 |
38 |
37 \item Given the following deterministic finite automata over the alphabet |
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38 $\{a, b\}$, find an automaton that recognises the complement language. |
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39 (Hint: Recall that for the algorithm from the lectures, the automaton needs to be |
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40 in completed form, that is have a transition for every letter from the alphabet.) |
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41 |
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42 \begin{center} |
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43 \begin{tikzpicture}[scale=3, line width=0.7mm] |
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44 \node[state, initial] (q0) at ( 0,1) {$q_0$}; |
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45 \node[state, accepting] (q1) at ( 1,1) {$q_1$}; |
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46 \path[->] (q0) edge node[above] {$a$} (q1) |
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47 (q1) edge [loop right] node {$b$} () |
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48 ; |
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49 \end{tikzpicture} |
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50 \end{center} |
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51 |
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52 \item Given the following deterministic finite automaton |
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53 |
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54 \begin{center} |
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55 \begin{tikzpicture}[scale=3, line width=0.7mm] |
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56 \node[state, initial] (q0) at ( 0,1) {$q_0$}; |
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57 \node[state,accepting] (q1) at ( 1,1) {$q_1$}; |
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58 \node[state, accepting] (q2) at ( 2,1) {$q_2$}; |
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59 \path[->] (q0) edge node[above] {$b$} (q1) |
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60 (q1) edge [loop above] node[above] {$a$} () |
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61 (q2) edge [loop above] node[above] {$a, b$} () |
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62 (q1) edge node[above] {$b$} (q2) |
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63 (q0) edge[bend right] node[below] {$a$} (q2) |
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64 ; |
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65 \end{tikzpicture} |
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66 \end{center} |
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67 find the corresponding minimal automaton. State clearly which nodes |
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68 can be merged. |
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69 |
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70 \item Given the following non-deterministic finite automaton over the alphabet $\{a, b\}$, |
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71 find a deterministic finite automaton that recognises the same language: |
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72 |
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73 \begin{center} |
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74 \begin{tikzpicture}[scale=3, line width=0.7mm] |
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75 \node[state, initial] (q0) at ( 0,1) {$q_0$}; |
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76 \node[state] (q1) at ( 1,1) {$q_1$}; |
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77 \node[state, accepting] (q2) at ( 2,1) {$q_2$}; |
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78 \path[->] (q0) edge node[above] {$a$} (q1) |
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79 (q0) edge [loop above] node[above] {$b$} () |
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80 (q0) edge [loop below] node[below] {$a$} () |
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81 (q1) edge node[above] {$a$} (q2) |
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82 ; |
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83 \end{tikzpicture} |
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84 \end{center} |
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85 |
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86 \item |
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87 Given the following finite deterministic automaton over the alphabet $\{a, b\}$: |
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88 |
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89 \begin{center} |
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90 \begin{tikzpicture}[scale=2, line width=0.5mm] |
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91 \node[state, initial, accepting] (q0) at ( 0,1) {$q_0$}; |
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92 \node[state, accepting] (q1) at ( 1,1) {$q_1$}; |
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93 \node[state] (q2) at ( 2,1) {$q_2$}; |
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94 \path[->] (q0) edge[bend left] node[above] {$a$} (q1) |
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95 (q1) edge[bend left] node[above] {$b$} (q0) |
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96 (q2) edge[bend left=50] node[below] {$b$} (q0) |
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97 (q1) edge node[above] {$a$} (q2) |
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98 (q2) edge [loop right] node {$a$} () |
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99 (q0) edge [loop below] node {$b$} () |
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100 ; |
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101 \end{tikzpicture} |
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102 \end{center} |
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103 |
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104 Give a regular expression that can recognise the same language as |
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105 this automaton. (Hint: If you use Brzozwski's method, you can assume |
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106 Arden's lemma which states that an equation of the form $q = q\cdot r + s$ |
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107 has the unique solution $q = s \cdot r^*$.)\ |
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108 |
39 |
109 \item Recall the definitions for $Der$ and $der$ from the lectures. |
40 \item Recall the definitions for $Der$ and $der$ from the lectures. |
110 Prove by induction on $r$ the property that |
41 Prove by induction on $r$ the property that |
111 |
42 |
112 \[ |
43 \[ |