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330 lines
9.0 KiB
TeX
330 lines
9.0 KiB
TeX
\documentclass{report}
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\usepackage{amsmath}
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\usepackage{tikz}
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\usetikzlibrary{automata, arrows.meta, positioning}
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\setcounter{chapter}{+1}
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\title{Language and Automata, Assignment 1}
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\author{Krzysztof Rudnicki\\ Student number: 307585}
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\date{\today}
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\begin{document}
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\maketitle
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\section{Regular expression}
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We are given following regular expression:
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\[ a^* + ba^*b + bba^*\]
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\section{Examples of accepted strings}
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\begin{enumerate}
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\item $ \varepsilon $
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\item a
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\item bab
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\item bba
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\item bb
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\end{enumerate}
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\section{Building NFA using Thompson construction algorithm}
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\begin{figure}
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\centering
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\begin{tikzpicture} [node distance = 2cm, on grid, auto]
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\node (q0) [state, initial, initial text = {}] {$q_0$};
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\node (q1) [state, accepting, right = of q0] {$q_1$};
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\path [-stealth, thick]
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(q0) edge node {$\varepsilon$} (q1);
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\end{tikzpicture}
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\caption{Operator 'a'} \label{fig:operatorA}
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\end{figure}
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\begin{figure}
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\centering
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\begin{tikzpicture} [node distance = 2cm, on grid, auto]
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\node (q0) [state, initial, initial text = {}] {$q_2$};
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\node (q1) [state, accepting, right = of q0] {$q_3$};
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\path [-stealth, thick]
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(q0) edge node {$\varepsilon$} (q1);
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\end{tikzpicture}
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\caption{Operator 'b'} \label{fig:operatorB}
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\end{figure}
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\begin{figure}
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\centering
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\begin{tikzpicture} [node distance = 2cm, on grid, auto]
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\node (q0) [state, initial, initial text = {}] {$q_4$};
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\node (q1) [state, right = of q0] {$q_0$};
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\node (q2) [state, right = of q1] {$q_1$};
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\node (q3) [state, accepting, right = of q2] {$q_5$};
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\path [-stealth, thick]
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(q0) edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q0) edge [bend right] node[below left] {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q1) edge node {a} (q2);
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\path [-stealth, thick]
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(q2) [bend left] edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q2) edge node {$\varepsilon$} (q3);
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\end{tikzpicture}
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\caption{Operator '$a^*$'} \label{fig:operatorAstar}
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\end{figure}
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\begin{figure}
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\centering
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\begin{tikzpicture} [node distance = 2cm, on grid, auto]
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\node (q6) [state, initial, initial text = {}] {$q_6$};
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\node (q0) [state, right = of q6] {$q_4$};
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\node (q1) [state, right = of q0] {$q_0$};
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\node (q2) [state, right = of q1] {$q_1$};
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\node (q3) [state, accepting, right = of q2] {$q_5$};
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\path [-stealth, thick]
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(q6) edge node {$b$} (q0);
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\path [-stealth, thick]
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(q0) edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q0) edge [bend right] node[below left] {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q1) edge node {a} (q2);
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\path [-stealth, thick]
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(q2) [bend left] edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q2) edge node {$\varepsilon$} (q3);
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\end{tikzpicture}
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\caption{Operator '$ba^*$'} \label{fig:operatorbastar}
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\end{figure}
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\begin{figure}
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\centering
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\begin{tikzpicture} [node distance = 2cm, on grid, auto]
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\node (q6) [state, initial, initial text = {}] {$q_6$};
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\node (q0) [state, right = of q6] {$q_4$};
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\node (q1) [state, right = of q0] {$q_0$};
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\node (q2) [state, right = of q1] {$q_1$};
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\node (q3) [state, right = of q2] {$q_5$};
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\node (q7) [state, accepting, right = of q3] {$q_7$};
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\path [-stealth, thick]
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(q6) edge node {$b$} (q0);
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\path [-stealth, thick]
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(q0) edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q0) edge [bend right] node[below left] {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q1) edge node {a} (q2);
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\path [-stealth, thick]
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(q2) [bend left] edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q2) edge node {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q3) edge node {$b$} (q7);
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\end{tikzpicture}
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\caption{Operator '$ba^*b$'} \label{fig:operatorbastarb}
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\end{figure}
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\begin{figure}
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\centering
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\begin{tikzpicture} [node distance = 2cm, on grid, auto]
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\node (q8) [state, initial, initial text = {}] {$q_8$};
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\node (q9) [state, right = of q8] {$q_9$};
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\node (q4) [state, right = of q9] {$q_{4}$};
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\path [-stealth, thick]
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(q8) edge node {$b$} (q9);
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\path [-stealth, thick]
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(q9) edge node {$b$} (q4);
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\end{tikzpicture}
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\caption{Operator '$bb$'} \label{fig:operatorbb}
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\end{figure}
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\begin{figure}
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\centering
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\begin{tikzpicture} [node distance = 2cm, on grid, auto]
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\node (q8) [state, initial, initial text = {}] {$q_8$};
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\node (q9) [state, right = of q8] {$q_9$};
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\node (q0) [state, right = of q9] {$q_4$};
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\node (q1) [state, right = of q0] {$q_0$};
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\node (q2) [state, right = of q1] {$q_1$};
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\node (q3) [state, accepting, right = of q2] {$q_5$};
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\path [-stealth, thick]
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(q0) edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q0) edge [bend right] node[below left] {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q1) edge node {a} (q2);
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\path [-stealth, thick]
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(q2) [bend left] edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q2) edge node {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q8) edge node {$b$} (q9);
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\path [-stealth, thick]
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(q9) edge node {$b$} (q4);
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\end{tikzpicture}
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\caption{Operator '$bba^*$'} \label{fig:operatorbbastar}
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\end{figure}
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\begin{figure}
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\centering
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\begin{tikzpicture} [node distance = 2cm, on grid, auto]
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\node (q10) [state, initial, initial text = {}] {$q_{10}$};
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% a*
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\node (q0) [state, above right = of q10] {$q_4$};
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\node (q1) [state, right = of q0] {$q_0$};
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\node (q2) [state, right = of q1] {$q_1$};
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\node (q3) [state, right = of q2] {$q_5$};
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% ba*b
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\node (q4) [state, below right = of q10] {$q_6$};
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\node (q5) [state, right = of q4] {$q_4$};
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\node (q6) [state, right = of q5] {$q_0$};
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\node (q7) [state, right = of q6] {$q_1$};
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\node (q8) [state, right = of q7] {$q_5$};
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\node (q9) [state, right = of q8] {$q_7$};
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\node (q11) [state, accepting, above right = of q9] {$q_{11}$};
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\path [-stealth, thick]
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(q10) edge node {$\varepsilon$} (q0);
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\path [-stealth, thick]
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(q10) edge node {$\varepsilon$} (q4);
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% a*
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\path [-stealth, thick]
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(q0) edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q0) edge [bend right] node[below left] {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q1) edge node {a} (q2);
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\path [-stealth, thick]
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(q2) [bend left] edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q2) edge node {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q3) edge node {$\varepsilon$} (q11);
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% ba*b
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\path [-stealth, thick]
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(q4) edge node {$b$} (q5);
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\path [-stealth, thick]
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(q5) edge node {$\varepsilon$} (q6);
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\path [-stealth, thick]
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(q5) edge [bend right] node[below left] {$\varepsilon$} (q8);
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\path [-stealth, thick]
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(q6) edge node {a} (q7);
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\path [-stealth, thick]
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(q7) [bend left] edge node {$\varepsilon$} (q6);
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\path [-stealth, thick]
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(q7) edge node {$\varepsilon$} (q8);
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\path [-stealth, thick]
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(q8) edge node {$b$} (q9);
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\path [-stealth, thick]
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(q9) edge node {$\varepsilon$} (q11);
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\end{tikzpicture}
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\caption{Operator '$a^*+ba^*b$'} \label{fig:operatorastarOrbastarb}
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\end{figure}
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\begin{figure}
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\begin{tikzpicture} [node distance = 1.5cm, on grid, auto]
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\node (q12) [state, initial, initial text = {}] {$q_{12}$};
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\node (q10) [state, above right = of q12] {$q_{10}$};
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% a*
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\node (q0) [state, above right = of q10] {$q_4$};
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\node (q1) [state, right = of q0] {$q_0$};
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\node (q2) [state, right = of q1] {$q_1$};
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\node (q3) [state, right = of q2] {$q_5$};
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% ba*b
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\node (q4) [state, below right = of q10] {$q_6$};
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\node (q5) [state, right = of q4] {$q_4$};
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\node (q6) [state, right = of q5] {$q_0$};
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\node (q7) [state, right = of q6] {$q_1$};
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\node (q8) [state, right = of q7] {$q_5$};
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\node (q9) [state, right = of q8] {$q_7$};
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\node (q11) [state, above right = of q9] {$q_{11}$};
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\node (q13) [state, accepting, below right = of q11] {$q_{13}$};
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\path [-stealth, thick]
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(q12) edge node {$\varepsilon$} (q10);
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\path [-stealth, thick]
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(q10) edge node {$\varepsilon$} (q0);
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\path [-stealth, thick]
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(q10) edge node {$\varepsilon$} (q4);
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% a*
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\path [-stealth, thick]
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(q0) edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q0) edge [bend right] node[below left] {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q1) edge node {a} (q2);
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\path [-stealth, thick]
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(q2) [bend left] edge node {$\varepsilon$} (q1);
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\path [-stealth, thick]
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(q2) edge node {$\varepsilon$} (q3);
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\path [-stealth, thick]
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(q3) edge node {$\varepsilon$} (q11);
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% ba*b
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\path [-stealth, thick]
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(q4) edge node {$b$} (q5);
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\path [-stealth, thick]
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(q5) edge node {$\varepsilon$} (q6);
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\path [-stealth, thick]
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(q5) edge [bend right] node[below left] {$\varepsilon$} (q8);
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\path [-stealth, thick]
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(q6) edge node {a} (q7);
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\path [-stealth, thick]
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(q7) [bend left] edge node {$\varepsilon$} (q6);
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\path [-stealth, thick]
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(q7) edge node {$\varepsilon$} (q8);
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\path [-stealth, thick]
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(q8) edge node {$b$} (q9);
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\path [-stealth, thick]
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(q9) edge node {$\varepsilon$} (q11);
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\path [-stealth, thick]
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(q11) edge node {$\varepsilon$} (q13);
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% bba*
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\end{tikzpicture}
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\caption{Operator '$a^*+ba^*b + bba^*$'} \label{fig:final}
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\end{figure}
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\section{Transforming NFA into DFA using subset algorithm}
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\section{Constructing minimal state DFA}
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\end{document}
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