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positions out of 5000i,500n,10000p,200000b,80000s + +Output written on project.pdf (4 pages, 86781 bytes). +PDF statistics: + 48 PDF objects out of 1000 (max. 8388607) + 34 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 13 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/project.pdf b/project.pdf new file mode 100644 index 00000000..d121581e Binary files /dev/null and b/project.pdf differ diff --git a/project.synctex.gz b/project.synctex.gz new file mode 100644 index 00000000..26f3e418 Binary files /dev/null and b/project.synctex.gz differ diff --git a/project.tex b/project.tex new file mode 100644 index 00000000..4d39c974 --- /dev/null +++ b/project.tex @@ -0,0 +1,329 @@ +\documentclass{report} +\usepackage{amsmath} +\usepackage{tikz} +\usetikzlibrary{automata, 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} +\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} +\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} +\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} +\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, 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} +\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} +\centering +\begin{tikzpicture} [node distance = 2cm, on grid, auto] + +\node (q8) [state, initial, initial text = {}] {$q_8$}; +\node (q9) [state, right = of q8] {$q_9$}; +\node (q4) [state, 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} +\centering +\begin{tikzpicture} [node distance = 2cm, on grid, auto] + +\node (q8) [state, initial, initial text = {}] {$q_8$}; +\node (q9) [state, right = of q8] {$q_9$}; +\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} +\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] {$q_4$}; +\node (q6) [state, right = of q5] {$q_0$}; +\node (q7) [state, right = of q6] {$q_1$}; +\node (q8) [state, right = of q7] {$q_5$}; +\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} +\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] {$q_4$}; +\node (q6) [state, right = of q5] {$q_0$}; +\node (q7) [state, right = of q6] {$q_1$}; +\node (q8) [state, right = of q7] {$q_5$}; +\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] {$q_{13}$}; + + +\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 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); +\path [-stealth, thick] + (q11) edge node {$\varepsilon$} (q13); + +% bba* + +\end{tikzpicture} +\caption{Operator '$a^*+ba^*b + bba^*$'} \label{fig:final} +\end{figure} + + +\section{Transforming NFA into DFA using subset algorithm} +\section{Constructing minimal state DFA} + + + +\end{document}