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PolishPigeon 2021-11-17 04:59:30 +01:00
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7 changed files with 120 additions and 29 deletions
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elacProjectA/project.aux
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\documentclass{report}
\documentclass{report}[pt12]
\usepackage{amsmath}
\usepackage{tikz}
\usepackage[section]{placeins}
@ -444,10 +444,91 @@ We are given following regular expression:
% bba*
\end{tikzpicture}
\caption{Operator '$a^*+ba^*b + bba^*$' - changed names of states} \label{fig:final}
\caption{Operator '$a^*+ba^*b + bba^*$' - changed names of states} \label{fig:finalBetter}
\end{figure}
\section{Transforming NFA into DFA using subset algorithm}
I will use $\epsilon_{cl}$ instead of $\epsilon$-closure for brevity sake.
Final state - $\underline{q_{19}}$ was marked with an $\underline{underline}$ and so did all the states of DFA that contain it.
\[ A = \epsilon_{cl}(q_0) = (q_0, q_1, q_3, q_4, q_6, q_{12}, q_{18}, \underline{q_{19}}) = \underline{A} \]
\[ \epsilon_{cl}(move(\underline{A}, a)) = (\epsilon_{cl}(move(q_0, q_1, q_3, q_4, q_6, q_{12}, q_{18}, \underline{q_{19}}, a))) = \epsilon_{cl}(q_9) = (q_9, q_{12}, q_{18}, \underline{q_{19}}) = \underline{B} \]
\[ A = \epsilon_{cl}(move(\underline{A}, b)) = \epsilon_{cl}(move(q_0, q_1, q_3, q_4, q_6, q_{12}, q_{18}, \underline{q_{19}}, b))) = \epsilon_{cl}(q_2, q_7) = (q_{10}, q_{15}) = C \]
\[ \epsilon_{cl}(move(\underline{B}, a)) = \epsilon_{cl}(move(q_9, q_{12}, q_{18}, \underline{q_{19}}), a) = \emptyset \]
\[ \epsilon_{cl}(move(\underline{B}, b)) = \epsilon_{cl}(move(q_9, q_{12}, q_{18}, \underline{q_{19}}), b) = \emptyset \]
\[ \epsilon_{cl}(move(C, a)) = \epsilon_{cl}(move(q_{10}, q_{15}), a) = \epsilon_{cl}(q_{13}) = (q_{15}) = D \]
\[ \epsilon_{cl}(move(C, b)) = \epsilon_{cl}(move(q_{10}, q_{15}), b) = \epsilon_{cl}(q_{17}) = (q_{18}, \underline{q_{19}}) = \underline{E} \]
\[ \epsilon_{cl}(move(D, a)) = \epsilon_{cl}(move((q_{15}), a) = \epsilon_{cl}(\emptyset) = \emptyset \]
\[ \epsilon_{cl}(move(D, b)) = \epsilon_{cl}(move((q_{15}), b) = \epsilon_{cl}(q_{17}) = (q_{18}, \underline{q_{19}}) = \underline{E} \]
\[ \epsilon_{cl}(move(\underline{E}, a)) = \epsilon_{cl}(move((q_{18}, \underline{q_{19}}), a) = \epsilon_{cl}(\emptyset) = \emptyset \]
\[ \epsilon_{cl}(move(\underline{E}, b)) = \epsilon_{cl}(move((q_{18}, \underline{q_{19}}), b) = \epsilon_{cl}(\emptyset) = \emptyset \]
\subsection{State table}
\begin{center}
\begin{tabular}{||c c c||}
\hline
State & a & b \\
\hline
$\underline{A}$ & $\underline{B}$ & C\\
\hline
$\underline{B}$ & $\emptyset$ & $\emptyset$ \\
\hline
C & D & $\underline{E}$\\
\hline
D & $\emptyset$ & $\underline{E}$\\
\hline
$\underline{E}$ & $\emptyset$ & $\emptyset$\\
\hline
$\emptyset$ & $\emptyset$ & $\emptyset$\\
\hline
\end{tabular}
\end{center}
\begin{figure}[!htb]
\centering
\begin{tikzpicture} [node distance = 2cm, on grid, auto]
\node (q0) [state, accepting, initial, initial text = {}] {A};
\node (q1) [state, accepting, above right = of q0] {B};
\node (q2) [state, below right = of q0] {C};
\node (q3) [state, accepting, above right = of q2] {E};
\node (q4) [state, below right = of q2] {D};
\node (q5) [state, right = of q1] {$\emptyset$};
\path [-stealth, thick]
(q0) edge node {$a$} (q1);
\path [-stealth, thick]
(q0) edge node {$b$} (q2);
\path [-stealth, thick]
(q2) edge node {$a$} (q4);
\path [-stealth, thick]
(q2) edge node {$b$} (q3);
\path [-stealth, thick]
(q4) edge node {$b$} (q3);
\path [-stealth, thick]
(q1) edge [bend left] node {$a, b$} (q5);
\path [-stealth, thick]
(q3) edge node {$a, b$} (q5);
\path [-stealth, thick]
(q4) edge [bend right] node {$a$} (q5);
\path [-stealth, thick]
(q5) edge [loop] node {$a, b$} (q5);
\end{tikzpicture}
\caption{DFA graph before minimalization} \label{dfaNominimalization}
\end{figure}
\section{Constructing minimal state DFA}