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1) Let the event set be $ E=\{a, b, g\} $, and consider the two languages $ L_{1}=\{\varepsilon, a, a b g\} $, and $ L_{2}=\{b\} $ c) $ \frac{L_{1}}{} L_{2}= $ ? d) $ L_{1}^{*}= $ ? 2) Let the event set be $ E=\{a, b, g\} $. Consider two state transition diagrams of automata are depicted in Figure 2.(a) and 2.(b), where nodes represent states and labeled arcs represent transitions between these states. Drawing the result of the product of two automata. (a). Automata $ G_{1} $ (b). Automata $ G_{2} $ Figure 2. Automata $ G_{1} $ and $ G_{2} $ 3) The two automata $ G $ and $ H_{a} $ are shown in Figure 3. Take $ M=\mathcal{L}(G) $ and $ K=\mathcal{L}_{m}\left(H_{a}\right) $. c) Form the product $ H_{a} \times G $ d) If $ E_{u c} $ is a set of uncontrollable events $ E_{u c}=\left\{b_{2}\right\} . K $ is controllable? (a). Automata $ G $ (b). Automata $ H_{a} $ Figure 3. Automata $ G $ and $ H_{a} $ 4) The two automata $ G $ and $ H_{a} $ are shown in Figure 3. Take $ M=\mathcal{L}(G) $ and $ K=\mathcal{L}_{m}\left(H_{a}\right) $. Let the set of uncontrollable events be $ E_{u c}=\left\{b_{2}\right\} $. c) Showing $ K $ as the set of string d) Find the Supremal Controllable Sublanguage of $ K: K^{\uparrow} C $

          1) Let the event set be \( E=\{a, b, g\} \), and consider the two languages \( L_{1}=\{\varepsilon, a, a b g\} \), and \( L_{2}=\{b\} \)
c) \( \frac{L_{1}}{} L_{2}= \) ?
d) \( L_{1}^{*}= \) ?
2) Let the event set be \( E=\{a, b, g\} \). Consider two state transition diagrams of automata are depicted in Figure 2.(a) and 2.(b), where nodes represent states and labeled arcs represent transitions between these states.
Drawing the result of the product of two automata.
(a). Automata \( G_{1} \)
(b). Automata \( G_{2} \)
Figure 2. Automata \( G_{1} \) and \( G_{2} \)
3) The two automata \( G \) and \( H_{a} \) are shown in Figure 3. Take \( M=\mathcal{L}(G) \) and \( K=\mathcal{L}_{m}\left(H_{a}\right) \).
c) Form the product \( H_{a} \times G \)
d) If \( E_{u c} \) is a set of uncontrollable events \( E_{u c}=\left\{b_{2}\right\} . K \) is controllable?
(a). Automata \( G \)
(b). Automata \( H_{a} \)
Figure 3. Automata \( G \) and \( H_{a} \)
4) The two automata \( G \) and \( H_{a} \) are shown in Figure 3. Take \( M=\mathcal{L}(G) \) and \( K=\mathcal{L}_{m}\left(H_{a}\right) \). Let the set of uncontrollable events be \( E_{u c}=\left\{b_{2}\right\} \).
c) Showing \( K \) as the set of string
d) Find the Supremal Controllable Sublanguage of \( K: K^{\uparrow} C \)

$1) Let the event set be E={a, b, g}, and consider the two languages L1={ε, a, a b g}, and L2={b} c) (L1)/() L2= ? d) L1^*= ? 2) Let the event set be E={a, b, g}. Consider two state transition diagrams of automata are depicted in Figure 2.(a) and 2.(b), where nodes represent states and labeled arcs represent transitions between these states. Drawing the result of the product of two automata. (a). Automata G1 (b). Automata G2 Figure 2. Automata G1 and G2 3) The two automata G and Ha are shown in Figure 3. Take M=ℒ(G) and K=ℒm(Ha). c) Form the product Ha× G d) If Eu c is a set of uncontrollable events Eu c={b2} . K is controllable? (a). Automata G (b). Automata Ha Figure 3. Automata G and Ha 4) The two automata G and Ha are shown in Figure 3. Take M=ℒ(G) and K=ℒm(Ha). Let the set of uncontrollable events be Eu c={b2}. c) Showing K as the set of string d) Find the Supremal Controllable Sublanguage of K: K^↑ C$

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