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Michael Sipser Edition 3 Exercise 1 Question 11 (Page No. 85)
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Prove that every $\text{NFA}$ can be converted to an equivalent one that has a single accept state.
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From every final state of the NFA we can add an Ɛ-transition to a new state and make it the final state, and make all the previous final states to non final states. Hopefully this'd work.
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Michael Sipser Edition 3 Exercise 1 Question 72 (Page No. 93)
Let $M_{1}$ and $M_{2}$ be $\text{DFA's}$ that have $k_{1}$ and $k_{2}$ states, respectively, and then let $U = L(M_{1})\cup L(M_{2}).$ Show that if $U\neq\phi$ then $U$ contains some string $s,$ where $|s| < max(k1, k2).$ Show that if $U\neq\sum^{*},$ then $U$ excludes some string $s,$ where $|s| < k1k2.$
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Michael Sipser Edition 3 Exercise 1 Question 71 (Page No. 93)
Let $\sum = \{0,1\}$ Let $A=\{0^{k}u0^{k}|k\geq 1$ $\text{and}$ $u\in \sum^{*}\}.$ Show that $A$ is regular. Let $B=\{0^{k}1u0^{k}|k\geq 1$ $\text{and}$ $u\in \sum^{*}\}.$Show that $B$ is not regular.
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Michael Sipser Edition 3 Exercise 1 Question 70 (Page No. 93)
We define the $\text{avoids}$ operation for languages $A$ and $B$ to be $\text{A avoids B = {w| w ∈ A and w doesn’t contain any string in B as a substring}.}$ Prove that the class of regular languages is closed under the ${avoids}$ operation.
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Michael Sipser Edition 3 Exercise 1 Question 69 (Page No. 93)
Let $\sum=\{0,1\}.$ Let $WW_{k}=\{ww|w\in \sum^{*}$ and $w$ is of length $k\}.$ Show that for each $k,$ no DFA can recognize $WW_{k}$ with fewer than $2^{k}$ states. Describe a much smaller $NFA$ for $\overline{WW_{k}},$ the complement of $WW_{k}.$
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