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\author{Elizabeth Hunt}
\date{\today}
\title{HW 02}
\hypersetup{
 pdfauthor={Elizabeth Hunt},
 pdftitle={HW 02},
 pdfkeywords={},
 pdfsubject={},
 pdfcreator={Emacs 28.2 (Org mode 9.7-pre)}, 
 pdflang={English}}
\begin{document}

\maketitle
\setlength\parindent{0pt}

\section{Question One}
\label{sec:org859695d}
\subsection{Partition Refinement}
\label{sec:org55dda6e}
\{ \{q\textsubscript{0}, q\textsubscript{1}, q\textsubscript{3}\}, \{q\textsubscript{2}, q\textsubscript{4}\} \}

S\textsubscript{1} = \{(q\textsubscript{0}, q\textsubscript{1}), (q\textsubscript{0}, q\textsubscript{3}), (q\textsubscript{1}, q\textsubscript{3})\}
S\textsubscript{2} = \{(q\textsubscript{2}, q\textsubscript{4})\}

\(\delta\)(q\textsubscript{0}, 1) = q\textsubscript{3} \(\in\) S\textsubscript{1}
\(\delta\)(q\textsubscript{1}, 1) = q\textsubscript{4} \(\in\) S\textsubscript{2}

(q\textsubscript{0}, q\textsubscript{1}) need to be split

\(\delta\)(q\textsubscript{0}, 0) = q\textsubscript{1} \(\in\) S\textsubscript{1}
\(\delta\)(q\textsubscript{3}, 0) = q\textsubscript{2} \(\in\) S\textsubscript{2}

(q\textsubscript{1}, q\textsubscript{2}) need to be split

\(\forall\) x \(\in\) \(\Sigma\), \(\delta\)(q\textsubscript{1}, x) = \(\delta\)(q\textsubscript{3}, x)
so \{q\textsubscript{1}, q\textsubscript{3}\} does not need to be split

In S\textsubscript{2}, \(\delta\)(q\textsubscript{2}, 0) \(\in\) S\textsubscript{1} and \(\delta\)(q\textsubscript{4}, 0) \(\in\) S\textsubscript{2}, thus need to be split

Finally, the refined partitions are \{\{q\textsubscript{0}\}, \{q\textsubscript{1}, q\textsubscript{3}\}, \{q\textsubscript{2}\}, \{q\textsubscript{4}\}\}

\subsection{Minimization}
\label{sec:org870d4d3}
\begin{center}
\begin{tabular}{rllll}
a \(\in\) \(\Sigma\) & \{q\textsubscript{0}\} & \{q\textsubscript{1}, q\textsubscript{3}\} & \{q\textsubscript{2}\} & \{q\textsubscript{4}\}\\[0pt]
0 & \{q\textsubscript{1}, q\textsubscript{3}\} & \{q\textsubscript{2}\} & \{q\textsubscript{1}, q\textsubscript{3}\} & \{q\textsubscript{4}\}\\[0pt]
1 & \{q\textsubscript{1}, q\textsubscript{3}\} & \{q\textsubscript{4}\} & \{q\textsubscript{4}\} & \{q\textsubscript{4}\}\\[0pt]
\end{tabular}
\end{center}

with d\textsubscript{0} = \{q\textsubscript{0}\}, d\textsubscript{1} = \{q\textsubscript{1}, q\textsubscript{3}\}, d\textsubscript{2} = \{q\textsubscript{2}\} and d\textsubscript{3} = \{q\textsubscript{4}\}

\begin{center}
\includegraphics[width=350px]{./img/min_dfa.png}
\end{center}

\section{Question Two}
\label{sec:orga1d2018}
See attached python
\end{document}