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Problem 3

The formal description of a DFA \(M\) is $\left(\left\\{q_{1}, q_{2}, q_{3}, q_{4}, q_{5}\right\\},\\{\mathrm{u}, \mathrm{d}\\}, \delta, q_{3},\left\\{q_{3}\right\\}\right)\(, where \)\delta$ is given by the following table. Give the state diagram of this machine. $$ \begin{array}{c|cc} & \mathrm{u} & \mathrm{d} \\ \hline q_{1} & q_{1} & q_{2} \\ q_{2} & q_{1} & q_{3} \\ q_{3} & q_{2} & q_{4} \\ q_{4} & q_{3} & q_{5} \\ q_{5} & q_{4} & q_{5} \end{array} $$

Expert verified

The state diagram for the given DFA M is as follows:
1. States: \(q_1\), \(q_2\), \(q_3\), \(q_4\), and \(q_5\)
2. Input symbols: \(u\) and \(d\)
3. Initial state: \(q_3\) (with an incoming arrow)
4. Final state: \(q_3\) (double circle)
5. Transitions: \(q_1 \xrightarrow{u} q_1\), \(q_1 \xrightarrow{d} q_2\), \(q_2 \xrightarrow{u} q_1\), \(q_2 \xrightarrow{d} q_3\), \(q_3 \xrightarrow{u} q_2\), \(q_3 \xrightarrow{d} q_4\), \(q_4 \xrightarrow{u} q_3\), \(q_4 \xrightarrow{d} q_5\), \(q_5 \xrightarrow{u} q_4\), and \(q_5 \xrightarrow{d} q_5\).
![State Diagram](https://i.imgur.com/CrHKSvM.png)

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Chapter 1

Let \(\Sigma=\\{0,1\\} .\) Let $W W_{k}=\left\\{w w \mid w \in \Sigma^{*}\right.\( and \)w\( is of length \)\left.k\right\\}$. a. Show that for each \(k\), no DFA can recognize \(W W_{k}\) with fewer than \(2^{k}\) states. b. Describe a much smaller NFA for \(\overline{W W}_{k}\), the complement of $W W_{k}$.

Chapter 1

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Chapter 1

Let \(A / B=\\{w \mid w x \in A\) for some \(x \in B\\} .\) Show that if \(A\) is regular and \(B\) is any language, then \(A / B\) is regular.

Chapter 1

Let \(\Sigma=\\{0,1\\}\) and let \(D=\\{w \mid w\) contains an equal number of occurrences of the substrings 01 and 10\(\\} .\) Thus \(101 \in D\) because 101 contains a single 01 and a single 10 , but $1010 \notin D\( because 1010 contains two 10 s and one 01 . Show that \)D$ is a regular language.

Chapter 1

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