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https://github.com/kuhyx/WUT_Computer_Science.git
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Finalizing project
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@ -27,4 +27,6 @@
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\@writefile{lof}{\contentsline {figure}{\numberline {1.11}{\ignorespaces DFA graph before minimalization}}{5}{}\protected@file@percent }
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\newlabel{dfaNominimalization}{{1.11}{5}}
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\@writefile{toc}{\contentsline {section}{\numberline {1.5}Constructing minimal state DFA}{5}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {1.12}{\ignorespaces DFA graph after minimalization}}{6}{}\protected@file@percent }
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\newlabel{dfaMinimalization}{{1.12}{6}}
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\gdef \@abspage@last{7}
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@ -1,5 +1,5 @@
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# Fdb version 3
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"project.tex" 1637122831 19066 b12925272f2edf5b24213d5833ad0f56 ""
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(generated)
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"project.log"
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"project.pdf"
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"project.aux"
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"project.log"
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18
project.log
18
project.log
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This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2021.10.23) 17 NOV 2021 04:58
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This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2021.10.23) 17 NOV 2021 05:20
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entering extended mode
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restricted \write18 enabled.
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file:line:error style messages enabled.
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@ -423,20 +423,20 @@ Overfull \hbox (58.9759pt too wide) detected at line 455
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[4
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] [5] (./project.aux) )
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] [5] [6] (./project.aux) )
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BIN
project.pdf
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project.pdf
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73
project.tex
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project.tex
@ -531,6 +531,79 @@ $\emptyset$ & $\emptyset$ & $\emptyset$\\
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\section{Constructing minimal state DFA}
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\begin{tabular}{|*{7}{c|}}
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\cline{1-1}
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$\underline{A}$ \\ \cline{1-2}
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$\underline{B}$ & $x_1$ \\ \cline{1-3}
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C & $x_1$ & $x_1$ \\ \cline{1-4}
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D & $x_1$ & $x_1$ & $x_2$ \\ \cline{1-5}
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$\underline{E}$ & $x_1$ & $x_1$ & $x_1$ & $x_1$ \\ \cline{1-6}
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$\emptyset$ & $x_1$ & $x_1$ & $x_2$ & $x_2$ & $x_1$ \\ \hline
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& $\underline{A}$ & $\underline{B}$ & C & D & $\underline{E}$ & $\emptyset$ \\ \hline
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\end{tabular}
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\begin{enumerate}
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\item First I marked (with $x_1$) all the pairs in which at least one of them were final state:
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\[ ([\underline{A}, \emptyset], [\underline{A}, \underline{E}], [\underline{A}, D], [\underline{A}, C], [\underline{A}, \underline{B}]) \]
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\[ ([\underline{B}, \emptyset], ([\underline{B}, \underline{E}], ([\underline{B}, D], ([\underline{B}, C]) \]
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\[ ([\underline{E}, \emptyset], [\underline{E}, C], [\underline{E}, D])\]
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\item We are left with the pairs:
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\[ ([\emptyset , C], [\emptyset , D], [D , C] ) \]
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For pair: $ [\emptyset , C] $ C goes to final state $\underline{E}$ on transition 'b' therefore we mark it with $x_2$
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For pair: $ [\emptyset , D] $ D goes to final state $\underline{E}$ on transition 'b' therefore we mark it with $x_2$
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For pair: $ [D , C] $ both C and D go to final state $\underline{E}$ on transition 'b' therefore we mark it with $x_2$
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\end{enumerate}
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No states could be minimized! Therefore our final minimal state DFA looks like this:
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\begin{figure}[!htb]
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\centering
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\begin{tikzpicture} [node distance = 2cm, on grid, auto]
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\node (q0) [state, accepting, initial, initial text = {}] {A};
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\node (q1) [state, accepting, above right = of q0] {B};
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\node (q2) [state, below right = of q0] {C};
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\node (q3) [state, accepting, above right = of q2] {E};
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\node (q4) [state, below right = of q2] {D};
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\node (q5) [state, right = of q1] {$\emptyset$};
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\path [-stealth, thick]
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(q0) edge node {$a$} (q1);
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\path [-stealth, thick]
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(q0) edge node {$b$} (q2);
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\path [-stealth, thick]
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(q2) edge node {$a$} (q4);
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\path [-stealth, thick]
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(q2) edge node {$b$} (q3);
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\path [-stealth, thick]
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(q4) edge node {$b$} (q3);
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\path [-stealth, thick]
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(q1) edge [bend left] node {$a, b$} (q5);
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\path [-stealth, thick]
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(q3) edge node {$a, b$} (q5);
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\path [-stealth, thick]
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(q4) edge [bend right] node {$a$} (q5);
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\path [-stealth, thick]
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(q5) edge [loop] node {$a, b$} (q5);
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\end{tikzpicture}
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\caption{DFA graph after minimalization} \label{dfaMinimalization}
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\end{figure}
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\end{document}
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