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\begin{large}
\centerline {\bf Computer Science 423}
\centerline {\bf Fall 2024}
\centerline {\bf Homework 1}
\centerline{\bf My name}
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%\begin{Large}
\bf Due: beginning of class, Thursday, 9/12/2024
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Answer the following questions and submit typeset solutions by the due date.  As stated on the syllabus, any collaborators or 
outside sources must be listed under the corresponding problem.  Further, your final submission must be completely your own work.
\begin{enumerate}

\item {[3 points each]}  True or False.  No explanation necessary.
\begin{enumerate}
\item A regular language can be recognized by multiple deterministic finite automata.
\item The language recognized by a DFA with zero accept states is \{$\emptyset$\}.
\item For any language $A$, where  $\lvert A \rvert$  $>$ 0, \(A \circ A\) is a proper subset of $A^*$.

\end{enumerate}

Collaborators:

\item {[5 points each]} Draw DFA state diagrams (with four states and no crossing arcs) that accept the following languages with alphabet  $\Sigma=\{0,1\}$:
\begin{enumerate}
\item The set of strings with $011$ as a substring
\item The set of strings with $101$ as a substring
\item The set of strings with $111$ as a substring
\end{enumerate}

Collaborators:

\item {[6 points each]} Draw a DFA state diagram with three states that accepts language: 
\\ \hspace*{3cm}  $L=\{w\in \{0, 1\}^*\ | \ $ $w$ = 0$(10)^*$ $\}$ \\

Collaborators:

\item {[10 points each]} For each the following DFAs given in transition table format,
\begin{enumerate}
\item[i] draw the corresponding state diagram
\item[ii] write the 5-tuple formal definition (if any of the components represents a set, list the entire set; you do not have to expand $\delta$)
\item[iii] give a precise description of the language accepted.
\end{enumerate}

\begin{enumerate}

\item
\begin{tabular}{r||c|c}
& 0 & 1 \\\hline\hline
$\rightarrow$ $q_0$ & $q_1$ & $q_0$ \\\hline
$q_1$ & $q_2$ & $q_1$ \\\hline
*$q_2$ & $q_2$ & $q_2$ \\
\end{tabular}

\item
\begin{tabular}{r||c|c}
& 0 & 1\\\hline
$\rightarrow$ $q_0$ & $q_1$ & $q_3$ \\\hline
$q_1$ & $q_1$ & $q_2$ \\\hline
*$q_2$ & $q_4$ & $q_3$ \\\hline
$q_3$ & $q_4$ & $q_3$ \\\hline
*$q_4$ & $q_1$ & $q_2$ \\
\end{tabular}

\end{enumerate}

Collaborators:

\end{enumerate}

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