From 635891af2f493fd0e247f3d0a20945b6e5bb7ce1 Mon Sep 17 00:00:00 2001 From: PolishPigeon Date: Wed, 17 Nov 2021 05:22:35 +0100 Subject: [PATCH] Finalizing project --- project.aux | 4 ++- project.fdb_latexmk | 8 ++--- project.log | 18 +++++------ project.pdf | Bin 114865 -> 119887 bytes project.synctex.gz | Bin 28205 -> 34994 bytes project.tex | 73 ++++++++++++++++++++++++++++++++++++++++++++ 6 files changed, 89 insertions(+), 14 deletions(-) diff --git a/project.aux b/project.aux index f4346f0c..dab0405b 100644 --- a/project.aux +++ b/project.aux @@ -27,4 +27,6 @@ \@writefile{lof}{\contentsline {figure}{\numberline {1.11}{\ignorespaces DFA graph before minimalization}}{5}{}\protected@file@percent } \newlabel{dfaNominimalization}{{1.11}{5}} \@writefile{toc}{\contentsline {section}{\numberline {1.5}Constructing minimal state DFA}{5}{}\protected@file@percent } -\gdef \@abspage@last{6} +\@writefile{lof}{\contentsline {figure}{\numberline {1.12}{\ignorespaces DFA graph after minimalization}}{6}{}\protected@file@percent } +\newlabel{dfaMinimalization}{{1.12}{6}} +\gdef \@abspage@last{7} diff --git a/project.fdb_latexmk b/project.fdb_latexmk index 81895ed9..d82fcca2 100644 --- a/project.fdb_latexmk +++ b/project.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1637121514 "project.tex" "project.pdf" "project" 1637121515 +["pdflatex"] 1637122832 "project.tex" "project.pdf" "project" 1637122833 "/etc/texmf/web2c/texmf.cnf" 1635008344 475 c0e671620eb5563b2130f56340a5fde8 "" "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" @@ -125,9 +125,9 @@ "/usr/share/texmf/web2c/texmf.cnf" 1613593815 38841 799d1dd9682a55ce442e10c99777ecc1 "" "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1635008389 5160710 ecf427ae8fa19139d8691f526e47bb9b "" "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1635008460 2570450 6e12b1c097cbda0f70015645294afd24 "" - "project.aux" 1637121515 2592 65a2c5d3cc5d4fbb767d5e48f951d8a4 "pdflatex" - "project.tex" 1637121514 16514 c0b41610a76b6798be6eeebc2a81d40a "" + "project.aux" 1637122833 2768 ba7afbf6aa8cf2ecf75244c6b0b5947d "pdflatex" + "project.tex" 1637122831 19066 b12925272f2edf5b24213d5833ad0f56 "" (generated) + "project.log" "project.pdf" "project.aux" - "project.log" diff --git a/project.log b/project.log index fc0e76e7..a83936a1 100644 --- a/project.log +++ b/project.log @@ -1,4 +1,4 @@ -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 +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 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -423,20 +423,20 @@ Overfull \hbox (58.9759pt too wide) detected at line 455 [4 -] [5] (./project.aux) ) +] [5] [6] (./project.aux) ) Here is how much of TeX's memory you used: - 13908 strings out of 479304 - 290559 string characters out of 5869778 - 665530 words of memory out of 5000000 - 30853 multiletter control sequences out of 15000+600000 + 13909 strings out of 479304 + 290578 string characters out of 5869778 + 665541 words of memory out of 5000000 + 30854 multiletter control sequences out of 15000+600000 408505 words of font info for 45 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 113i,10n,116p,418b,949s stack positions out of 5000i,500n,10000p,200000b,80000s -Output written on project.pdf (6 pages, 114865 bytes). +Output written on project.pdf (7 pages, 119887 bytes). 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$x_1$ & $x_1$ & $x_2$ & $x_2$ & $x_1$ \\ \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] +(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 after minimalization} \label{dfaMinimalization} +\end{figure} + + \end{document}