From 79e25b89857f91728cb8906ddf6b86c26051618e Mon Sep 17 00:00:00 2001 From: PolishPigeon Date: Fri, 17 Jun 2022 23:55:59 +0200 Subject: [PATCH] chore --- README.md | 6 - project.aux | 32 --- project.fdb_latexmk | 133 --------- project.fls | 145 ---------- project.log | 0 project.pdf | Bin 120280 -> 0 bytes project.synctex(busy) | 0 project.synctex.gz | Bin 36182 -> 0 bytes project.tex | 617 ------------------------------------------ 9 files changed, 933 deletions(-) delete mode 100644 README.md delete mode 100644 project.aux delete mode 100644 project.fdb_latexmk delete mode 100644 project.fls delete mode 100644 project.log delete mode 100644 project.pdf delete mode 100644 project.synctex(busy) delete mode 100644 project.synctex.gz delete mode 100644 project.tex diff --git a/README.md b/README.md deleted file mode 100644 index 3a8db2bb..00000000 --- a/README.md +++ /dev/null @@ -1,6 +0,0 @@ -<<<<<<< HEAD -# WUT_Computer_Science -This repository aims to consolidate every single project I did solo during my Computer Science course on Warsaw University of Technology -======= -# elacProjectA ->>>>>>> elacProjectA/main diff --git a/project.aux b/project.aux deleted file mode 100644 index dab0405b..00000000 --- a/project.aux +++ /dev/null @@ -1,32 +0,0 @@ -\relax -\@writefile{toc}{\contentsline {section}{\numberline {1.1}Regular expression}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\numberline {1.2}Examples of accepted strings}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\numberline {1.3}Building NFA using Thompson construction algorithm}{1}{}\protected@file@percent } -\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces Operator 'a'}}{1}{}\protected@file@percent } -\newlabel{fig:operatorA}{{1.1}{1}} -\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces Operator 'b'}}{1}{}\protected@file@percent } -\newlabel{fig:operatorB}{{1.2}{1}} -\@writefile{lof}{\contentsline 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arrows.meta, positioning} - - - -\setcounter{chapter}{+1} - -\title{Language and Automata, Assignment 1} -\author{Krzysztof Rudnicki\\ Student number: 307585} -\date{\today} - - -\begin{document} -\maketitle - -\section{Regular expression} -We are given following regular expression: -\[ a^* + ba^*b + bba^*\] - -\section{Examples of accepted strings} -\begin{enumerate} -\item $ \varepsilon $ -\item a -\item bab -\item bba -\item bb -\end{enumerate} - -\section{Building NFA using Thompson construction algorithm} -\begin{figure}[!htb] -\centering -\begin{tikzpicture} [node distance = 2cm, on grid, auto] - -\node (q0) [state, initial, initial text = {}] {$q_0$}; -\node (q1) [state, accepting, right = of q0] {$q_1$}; - -\path [-stealth, thick] - (q0) edge node {$\varepsilon$} (q1); -\end{tikzpicture} -\caption{Operator 'a'} \label{fig:operatorA} -\end{figure} - -\begin{figure}[!htb] -\centering -\begin{tikzpicture} [node distance = 2cm, on grid, auto] - -\node (q0) [state, initial, initial text = {}] {$q_2$}; -\node (q1) [state, accepting, right = of q0] {$q_3$}; - -\path [-stealth, thick] - (q0) edge node {$\varepsilon$} (q1); -\end{tikzpicture} -\caption{Operator 'b'} \label{fig:operatorB} -\end{figure} - -\begin{figure}[!htb] -\centering -\begin{tikzpicture} [node distance = 2cm, on grid, auto] - -\node (q0) [state, initial, initial text = {}] {$q_4$}; -\node (q1) [state, right = of q0] {$q_0$}; -\node (q2) [state, right = of q1] {$q_1$}; -\node (q3) [state, accepting, right = of q2] {$q_5$}; - -\path [-stealth, thick] - (q0) edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q0) edge [bend right] node[below left] {$\varepsilon$} (q3); -\path [-stealth, thick] - (q1) edge node {a} (q2); -\path [-stealth, thick] - (q2) [bend left] edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q2) edge node {$\varepsilon$} (q3); -\end{tikzpicture} -\caption{Operator '$a^*$'} \label{fig:operatorAstar} -\end{figure} - -\begin{figure}[!htb] -\centering -\begin{tikzpicture} [node distance = 2cm, on grid, auto] - -\node (q6) [state, initial, initial text = {}] {$q_2$}; -\node (q0) [state, right = of q6] {$q_4$}; -\node (q1) [state, right = of q0] {$q_0$}; -\node (q2) [state, right = of q1] {$q_1$}; -\node (q3) [state, accepting, right = of q2] {$q_5$}; - -\path [-stealth, thick] - (q6) edge node {$b$} (q0); -\path [-stealth, thick] - (q0) edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q0) edge [bend right] node[below left] {$\varepsilon$} (q3); -\path [-stealth, thick] - (q1) edge node {a} (q2); -\path [-stealth, thick] - (q2) [bend left] edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q2) edge node {$\varepsilon$} (q3); -\end{tikzpicture} -\caption{Operator '$ba^*$'} \label{fig:operatorbastar} -\end{figure} - -\begin{figure}[!htb] -\centering -\begin{tikzpicture} [node distance = 2cm, on grid, auto] - -\node (q6) [state, initial, initial text = {}] {$q_6$}; -\node (q0) [state, right = of q6] {$q_4$}; -\node (q1) [state, right = of q0] {$q_0$}; -\node (q2) [state, right = of q1] {$q_1$}; -\node (q3) [state, right = of q2] {$q_5$}; -\node (q7) [state, accepting, right = of q3] {$q_7$}; - -\path [-stealth, thick] - (q6) edge node {$b$} (q0); -\path [-stealth, thick] - (q0) edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q0) edge [bend right] node[below left] {$\varepsilon$} (q3); -\path [-stealth, thick] - (q1) edge node {a} (q2); -\path [-stealth, thick] - (q2) [bend left] edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q2) edge node {$\varepsilon$} (q3); -\path [-stealth, thick] - (q3) edge node {$b$} (q7); - -\end{tikzpicture} -\caption{Operator '$ba^*b$'} \label{fig:operatorbastarb} -\end{figure} - -\begin{figure}[!htb] -\centering -\begin{tikzpicture} [node distance = 2cm, on grid, auto] - -\node (q8) [state, initial, initial text = {}] {$q_2$}; -\node (q9) [state, right = of q8] {$q_3$}; -\node (q4) [state, accepting, right = of q9] {$q_{4}$}; - -\path [-stealth, thick] - (q8) edge node {$b$} (q9); -\path [-stealth, thick] - (q9) edge node {$b$} (q4); - - -\end{tikzpicture} -\caption{Operator '$bb$'} \label{fig:operatorbb} -\end{figure} - -\begin{figure}[!htb] -\centering -\begin{tikzpicture} [node distance = 2cm, on grid, auto] - -\node (q8) [state, initial, initial text = {}] {$q_2$}; -\node (q9) [state, right = of q8] {$q_3$}; -\node (q0) [state, right = of q9] {$q_4$}; -\node (q1) [state, right = of q0] {$q_0$}; -\node (q2) [state, right = of q1] {$q_1$}; -\node (q3) [state, accepting, right = of q2] {$q_5$}; - -\path [-stealth, thick] - (q0) edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q0) edge [bend right] node[below left] {$\varepsilon$} (q3); -\path [-stealth, thick] - (q1) edge node {a} (q2); -\path [-stealth, thick] - (q2) [bend left] edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q2) edge node {$\varepsilon$} (q3); - -\path [-stealth, thick] - (q8) edge node {$b$} (q9); -\path [-stealth, thick] - (q9) edge node {$b$} (q4); -\end{tikzpicture} -\caption{Operator '$bba^*$'} \label{fig:operatorbbastar} -\end{figure} - -\begin{figure}[!htb] -\centering -\begin{tikzpicture} [node distance = 2cm, on grid, auto] - -\node (q10) [state, initial, initial text = {}] {$q_{10}$}; -% a* - -\node (q0) [state, above right = of q10] {$q_4$}; -\node (q1) [state, right = of q0] {$q_0$}; -\node (q2) [state, right = of q1] {$q_1$}; -\node (q3) [state, right = of q2] {$q_5$}; -% ba*b -\node (q4) [state, below right = of q10] {$q_6$}; -\node (q5) [state, right = of q4] {$q_8$}; -\node (q6) [state, right = of q5] {$q_9$}; -\node (q7) [state, right = of q6] {$q_{12}$}; -\node (q8) [state, right = of q7] {$q_{13}$}; -\node (q9) [state, right = of q8] {$q_7$}; - -\node (q11) [state, accepting, above right = of q9] {$q_{11}$}; - -\path [-stealth, thick] - (q10) edge node {$\varepsilon$} (q0); - -\path [-stealth, thick] - (q10) edge node {$\varepsilon$} (q4); - -% a* -\path [-stealth, thick] - (q0) edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q0) edge [bend right] node[below left] {$\varepsilon$} (q3); -\path [-stealth, thick] - (q1) edge node {a} (q2); -\path [-stealth, thick] - (q2) [bend left] edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q2) edge node {$\varepsilon$} (q3); -\path [-stealth, thick] - (q3) edge node {$\varepsilon$} (q11); - -% ba*b -\path [-stealth, thick] - (q4) edge node {$b$} (q5); -\path [-stealth, thick] - (q5) edge node {$\varepsilon$} (q6); -\path [-stealth, thick] - (q5) edge [bend right] node[below left] {$\varepsilon$} (q8); -\path [-stealth, thick] - (q6) edge node {a} (q7); -\path [-stealth, thick] - (q7) [bend left] edge node {$\varepsilon$} (q6); -\path [-stealth, thick] - (q7) edge node {$\varepsilon$} (q8); -\path [-stealth, thick] - (q8) edge node {$b$} (q9); -\path [-stealth, thick] - (q9) edge node {$\varepsilon$} (q11); -\end{tikzpicture} -\caption{Operator '$a^*+ba^*b$'} \label{fig:operatorastarOrbastarb} -\end{figure} - -\begin{figure}[!htb] -\begin{tikzpicture} [node distance = 1.5cm, on grid, auto] - -\node (q12) [state, initial, initial text = {}] {$q_{12}$}; - -\node (q10) [state, above right = of q12] {$q_{10}$}; -% a* - -\node (q0) [state, above right = of q10] {$q_4$}; -\node (q1) [state, right = of q0] {$q_0$}; -\node (q2) [state, right = of q1] {$q_1$}; -\node (q3) [state, right = of q2] {$q_5$}; -% ba*b -\node (q4) [state, below right = of q10] {$q_6$}; -\node (q5) [state, right = of q4] {$q_8$}; -\node (q6) [state, right = of q5] {$q_9$}; -\node (q7) [state, right = of q6] {$q_{12}$}; -\node (q8) [state, right = of q7] {$q_{13}$}; -\node (q9) [state, right = of q8] {$q_7$}; - -\node (q11) [state, above right = of q9] {$q_{11}$}; - -\node (q13) [state, accepting, below right = of q11] {$q_{13}$}; - -% bba* -\node (q14) [state, below right = of q12] {$q_{14}$}; -\node (q15) [state, right = of q14] {$q_{15}$}; -\node (q16) [state, right = of q15] {$q_{16}$}; -\node (q17) [state, right = of q16] {$q_{17}$}; -\node (q18) [state, right = of q17] {$q_{18}$}; -\node (q19) [state, right = of q18] {$q_{19}$}; - -\path [-stealth, thick] - (q12) edge node {$b$} (q14); - -\path [-stealth, thick] - (q14) edge node {$b$} (q15); -\path [-stealth, thick] - (q15) edge node {$b$} (q16); -\path [-stealth, thick] - (q16) edge node {$\varepsilon$} (q17); -\path [-stealth, thick] - (q16) edge [bend right] node[below left] {$\varepsilon$} (q19); -\path [-stealth, thick] - (q17) edge node {$a$} (q18); -\path [-stealth, thick] - (q18) edge node {$\varepsilon$} (q19); -\path [-stealth, thick] - (q18) [bend left] edge node {$\varepsilon$} (q17); -\path [-stealth, thick] - (q19) edge [bend right] node {$b$} (q13); - -\path [-stealth, thick] - (q12) edge node {$\varepsilon$} (q10); - -\path [-stealth, thick] - (q10) edge node {$\varepsilon$} (q0); - -\path [-stealth, thick] - (q10) edge node {$\varepsilon$} (q4); - -% a* -\path [-stealth, thick] - (q0) edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q0) edge [bend right] node[below left] {$\varepsilon$} (q3); -\path [-stealth, thick] - (q1) edge node {a} (q2); -\path [-stealth, thick] - (q2) [bend left] edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q2) edge node {$\varepsilon$} (q3); -\path [-stealth, thick] - (q3) edge node {$\varepsilon$} (q11); - -% ba*b -\path [-stealth, thick] - (q4) edge node {$b$} (q5); -\path [-stealth, thick] - (q5) edge node {$\varepsilon$} (q6); -\path [-stealth, thick] - (q5) edge [bend left] node[below left] {$\varepsilon$} (q8); -\path [-stealth, thick] - (q6) edge node {a} (q7); -\path [-stealth, thick] - (q7) [bend left] edge node {$\varepsilon$} (q6); -\path [-stealth, thick] - (q7) edge node {$\varepsilon$} (q8); -\path [-stealth, thick] - (q8) edge node {$b$} (q9); -\path [-stealth, thick] - (q9) edge node {$\varepsilon$} (q11); -\path [-stealth, thick] - (q11) edge node {$\varepsilon$} (q13); - -% bba* - -\end{tikzpicture} -\caption{Operator '$a^*+ba^*b + bba^*$'} \label{fig:final} -\end{figure} - -\begin{figure}[!htb] -\begin{tikzpicture} [node distance = 1.5cm, on grid, auto] - -\node (q12) [state, initial, initial text = {}] {$q_{0}$}; - -\node (q10) [state, above right = of q12] {$q_{1}$}; -% a* - -\node (q0) [state, above right = of q10] {$q_3$}; -\node (q1) [state, right = of q0] {$q_6$}; -\node (q2) [state, right = of q1] {$q_9$}; -\node (q3) [state, right = of q2] {$q_{12}$}; -% ba*b -\node (q4) [state, below right = of q10] {$q_4$}; -\node (q5) [state, right = of q4] {$q_7$}; -\node (q6) [state, right = of q5] {$q_{10}$}; -\node (q7) [state, right = of q6] {$q_{13}$}; -\node (q8) [state, right = of q7] {$q_{15}$}; -\node (q9) [state, right = of q8] {$q_{17}$}; - -\node (q11) [state, above right = of q9] {$q_{18}$}; - -\node (q13) [state, accepting, below right = of q11] {$q_{19}$}; - -% bba* -\node (q14) [state, below right = of q12] {$q_{2}$}; -\node (q15) [state, right = of q14] {$q_{5}$}; -\node (q16) [state, right = of q15] {$q_{8}$}; -\node (q17) [state, right = of q16] {$q_{11}$}; -\node (q18) [state, right = of q17] {$q_{14}$}; -\node (q19) [state, right = of q18] {$q_{16}$}; - -\path [-stealth, thick] - (q12) edge node {$b$} (q14); - -\path [-stealth, thick] - (q14) edge node {$b$} (q15); -\path [-stealth, thick] - (q15) edge node {$b$} (q16); -\path [-stealth, thick] - (q16) edge node {$\varepsilon$} (q17); -\path [-stealth, thick] - (q16) edge [bend right] node[below left] {$\varepsilon$} (q19); -\path [-stealth, thick] - (q17) edge node {$a$} (q18); -\path [-stealth, thick] - (q18) edge node {$\varepsilon$} (q19); -\path [-stealth, thick] - (q18) [bend left] edge node {$\varepsilon$} (q17); -\path [-stealth, thick] - (q19) edge [bend right] node {$b$} (q13); - -\path [-stealth, thick] - (q12) edge node {$\varepsilon$} (q10); - -\path [-stealth, thick] - (q10) edge node {$\varepsilon$} (q0); - -\path [-stealth, thick] - (q10) edge node {$\varepsilon$} (q4); - -% a* -\path [-stealth, thick] - (q0) edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q0) edge [bend right] node[below left] {$\varepsilon$} (q3); -\path [-stealth, thick] - (q1) edge node {a} (q2); -\path [-stealth, thick] - (q2) [bend left] edge node {$\varepsilon$} (q1); -\path [-stealth, thick] - (q2) edge node {$\varepsilon$} (q3); -\path [-stealth, thick] - (q3) edge node {$\varepsilon$} (q11); - -% ba*b -\path [-stealth, thick] - (q4) edge node {$b$} (q5); -\path [-stealth, thick] - (q5) edge node {$\varepsilon$} (q6); -\path [-stealth, thick] - (q5) edge [bend left] node[below left] {$\varepsilon$} (q8); -\path [-stealth, thick] - (q6) edge node {a} (q7); -\path [-stealth, thick] - (q7) [bend left] edge node {$\varepsilon$} (q6); -\path [-stealth, thick] - (q7) edge node {$\varepsilon$} (q8); -\path [-stealth, thick] - (q8) edge node {$b$} (q9); -\path [-stealth, thick] - (q9) edge node {$\varepsilon$} (q11); -\path [-stealth, thick] - (q11) edge node {$\varepsilon$} (q13); - -% bba* - -\end{tikzpicture} -\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. -\[ A = \epsilon_{cl}(q_0) = (q_0, q_1, q_3, q_4, q_6, q_{12}, q_{18}, \underline{q_{19}}) = \underline{A} \] -\[ \epsilon_{cl}(move(\underline{A}, a)) = (\epsilon_{cl}(move(q_0, q_1, q_3, q_4, q_6, q_{12}, q_{18}, \underline{q_{19}}, a))) = \epsilon_{cl}(q_9) = (q_6, q_9, q_{12}, q_{18}, \underline{q_{19}}) = \underline{B} \] -\[ A = \epsilon_{cl}(move(\underline{A}, b)) = \epsilon_{cl}(move(q_0, q_1, q_3, q_4, q_6, q_{12}, q_{18}, \underline{q_{19}}, b))) = \epsilon_{cl}(q_2, q_7) = (q_2, q_7, q_{10}, q_{15}) = C \] -\[ \epsilon_{cl}(move(\underline{B}, a)) = \epsilon_{cl}(move(q_6, q_9, q_{12}, q_{18}, \underline{q_{19}}), a) = \epsilon_{cl}(q_9) = (q_6, q_9, q_{12}, q_{18}, \underline{q_{19}}) = \underline{B} \] -\[ \epsilon_{cl}(move(\underline{B}, b)) = \epsilon_{cl}(move(q_6, q_9, q_{12}, q_{18}, \underline{q_{19}}), b) = \emptyset \] -\[ \epsilon_{cl}(move(C, a)) = \epsilon_{cl}(move(q_2, q_7, q_{10}, q_{15}), a) = \epsilon_{cl}(q_{13}) = (q_{10}, q_{13}, q_{15}) = D \] -\[ \epsilon_{cl}(move(C, b)) = \epsilon_{cl}(move((q_2, q_7, q_{10}, q_{15}), b) = \epsilon_{cl}(q_{17}) = (q_{17}, q_{18}, \underline{q_{19}}) = \underline{E} \] -\[ \epsilon_{cl}(move(D, a)) = \epsilon_{cl}(move((q_{10}, q_{13}, q_{15}), a) = \epsilon_{cl}(q_{13}) = (q_{10}, q_{13}, q_{15}) = D \] -\[ \epsilon_{cl}(move(D, b)) = \epsilon_{cl}(move(q_{10}, q_{13}, q_{15}, b) = \epsilon_{cl}(q_{17}) = (q_{17}, q_{18}, \underline{q_{19}}) = \underline{E} \] -\[ \epsilon_{cl}(move(\underline{E}, a)) = \epsilon_{cl}(move(q_{17}, q_{18}, \underline{q_{19}}), a) = \epsilon_{cl}(\emptyset) = \emptyset \] -\[ \epsilon_{cl}(move(\underline{E}, b)) = \epsilon_{cl}(move(q_{17}, q_{18}, \underline{q_{19}}), b) = \epsilon_{cl}(\emptyset) = \emptyset \] - -\subsection{State table} -\begin{center} -\begin{tabular}{||c c c||} - \hline - State & a & b \\ - \hline -$\underline{A}$ & $\underline{B}$ & C\\ - \hline - $\underline{B}$ & $\underline{B}$ & $\emptyset$ \\ - \hline - C & D & $\underline{E}$\\ - \hline - D & D & $\underline{E}$\\ - \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] - (q1) edge [loop] node {$a$} (q1); - - \path [-stealth, thick] - (q2) edge node {$b$} (q4); - - \path [-stealth, thick] - (q2) edge node {$b$} (q3); - - \path [-stealth, thick] - (q4) edge [bend right] node {$b$} (q3); - - \path [-stealth, thick] - (q4) edge [loop] node {$a$} (q4); - - \path [-stealth, thick] - (q1) edge [bend left] node {$a$} (q5); - - \path [-stealth, thick] - (q3) edge node {$a, b$} (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} - -\begin{tabular}{|*{7}{c|}} - \cline{1-1} - $\underline{A}$ \\ \cline{1-2} - $\underline{B}$ & $x_1$ \\ \cline{1-3} - C & $x_1$ & $x_1$ \\ \cline{1-4} - D & $x_1$ & $x_1$ & $x_2$ \\ \cline{1-5} - $\underline{E}$ & $x_1$ & $x_1$ & $x_1$ & $x_1$ \\ \cline{1-6} - $\emptyset$ & $x_1$ & $x_1$ & $x_2$ & $x_2$ & $x_1$ \\ \hline - & $\underline{A}$ & $\underline{B}$ & C & D & $\underline{E}$ & $\emptyset$ \\ \hline -\end{tabular} - -\begin{enumerate} - \item First I marked (with $x_1$) all the pairs in which at least one of them were final state: - \[ ([\underline{A}, \emptyset], [\underline{A}, \underline{E}], [\underline{A}, D], [\underline{A}, C], [\underline{A}, \underline{B}]) \] - \[ ([\underline{B}, \emptyset], ([\underline{B}, \underline{E}], ([\underline{B}, D], ([\underline{B}, C]) \] - \[ ([\underline{E}, \emptyset], [\underline{E}, C], [\underline{E}, D])\] - - \item We are left with the pairs: - \[ ([\emptyset , C], [\emptyset , D], [D , C] ) \] - For pair: $ [\emptyset , C] $ C goes to final state $\underline{E}$ on transition 'b' therefore we mark it with $x_2$ - For pair: $ [\emptyset , D] $ D goes to final state $\underline{E}$ on transition 'b' therefore we mark it with $x_2$ - For pair: $ [D , C] $ both C and D go to final state $\underline{E}$ on transition 'b' therefore we mark it with $x_2$ -\end{enumerate} - -No states could be minimized! Therefore our final minimal state DFA looks like this: - -\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] -(q1) edge [loop] node {$a$} (q1); - -\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 [below loop] node {$a$} (q3); - -\path [-stealth, thick] -(q3) edge node {$a$} (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 after minimalization} \label{dfaMinimalization} -\end{figure} - - - - -\end{document}