%--*- latex -*----------------------------------------------------------------- %$Author: saulius $ %$Date: 2020-06-04 14:58:32 +0300 (Thu, 04 Jun 2020) $ %$Revision: 1524 $ %$URL: svn+ssh://saulius-grazulis.lt/home/saulius/svn-repositories/seminarai/2020-verifikacjos-seminarui/slides.tex $ %------------------------------------------------------------------------------ \documentclass[mathserif]{beamer} \usetheme{Warwick} \useoutertheme{infolines} \setbeamertemplate{headline}{} % removes the headline the infolines inserts %\setbeamertemplate{footline}[frame number] \renewcommand\familydefault{\rmdefault} % For XeLaTeX: % https://tex.stackexchange.com/questions/452151/how-do-i-render-the-word-v%C7%ABlundarkvi%C3%B0a-with-bookman-and-xelatex % "Use an OpenType clone of Bookman, for instance TeX Gyre Bonum": \usepackage{fontspec} \setmainfont{TeX Gyre Bonum} \usepackage[style=authoryear,maxnames=1,doi=true,url=true,backend=biber]{biblatex} \addbibresource{bibliography/citations.bib} \newcommand{\mycite}{\parencite} \usepackage{multirow} \usepackage{colordvi} \usepackage{graphicx} \usepackage{tikz} \usetikzlibrary{matrix} \usepackage{verbatim} \usepackage{listings} \usepackage{chemfig} \usepackage{listings} % https://en.wikibooks.org/wiki/LaTeX/Algorithms % http://mirror.datacenter.by/pub/mirrors/CTAN/macros/latex/contrib/algorithmicx/algorithmicx.pdf \usepackage{algpseudocode} \usepackage{algorithm} \usepackage{amssymb} \include{commands} % https://tex.stackexchange.com/questions/595/how-can-i-get-bold-math-symbols: \newcommand{\boldm}[1]{\mathversion{bold}#1\mathversion{normal}} \newcommand{\RCSid}[1]{\fontsize{7pt}{7pt}\selectfont $#1$ \today} %%BEGIN LANGUAGE lt \title{Tranzistoriai ir loginiai elementai} %%END LANGUAGE lt \author{Saulius Gražulis} \date{Vilnius, 2020} \begin{document} %------------------------------------------------------------------------------ \input{text-parts/recording-announcement.tex} \begin{frame} \titlepage \input{affiliation_lt} \begin{center} \mbox{} \hfill\hfill\hfill \includegraphics[height=1.5cm]{images/sp_VU_zenklas.eps} \hfill \includegraphics[height=1.5cm]{images/2019-05-02_Melynas_MIF-zenklas242x244.png} \hfill\hfill\hfill \mbox{} \end{center} \vfill %% \tiny %% \RCSid{ %% $Id: slides.tex 1524 2020-06-04 11:58:32Z saulius $ %% } \begin{flushright} \begin{minipage}[c]{0.67\textwidth} \tiny\raggedright %%BEGIN LANGUAGE lt Šį skaidrių rinkinį galima kopijuoti, kaip nurodyta Creative Commons %%END LANGUAGE lt \myhref{http://creativecommons.org/licenses/by-sa/4.0/}{Attribution-ShareAlike 4.0 International} licenzijoje \end{minipage} %% \begin{minipage}[c]{1.5cm} \myhref{http://creativecommons.org/licenses/by-sa/4.0/}{ \includegraphics[width=1.5cm]{images/CC-BY-SA.eps} } \end{minipage} \end{flushright} \end{frame} %============================================================================== \begin{frame} \frametitle{Valdomas perjungiklis – tranzistorius} FET – Field-effect transistor MOSFET – Metal–oxide–semiconductor field-effect transistor \only<1>{ \begin{center} \hfill \parbox[c]{5cm}{ \begin{center} \includegraphics[height=5cm]{drawings/transistors/transistor-marked-termini_lt.eps} \end{center} } \hfill \parbox[c]{5cm}{ \begin{center} \includegraphics[height=5cm]{drawings/mechanical-switches/switch.eps} \end{center} } \hfill \mbox{} \end{center} } \only<2>{ \begin{center} \hfill \parbox[c]{5cm}{ \begin{center} \includegraphics[height=5cm]{drawings/transistors/transistor-marked-termini_lt.eps} \end{center} } \hfill \parbox[c]{5cm}{ \begin{center} \includegraphics[width=5cm]{images/Lateral_mosfet_lt.eps}\\ \vspace{\baselineskip} \footnotesize\em \myhref{https://commons.wikimedia.org/wiki/File:Lateral_mosfet.svg}{ Cyril Buttay} \myhref{https://creativecommons.org/licenses/by-sa/3.0/deed.en}{[CC BY-SA 3.0]} \end{center} } \hfill \mbox{} \end{center} } \only<3>{ \begin{center} \hfill \parbox[c]{5cm}{ \begin{center} \includegraphics[height=5cm]{drawings/transistors/transistor-marked-termini_lt.eps} \end{center} } \hfill \parbox[c]{5cm}{ \begin{center} \includegraphics[width=5cm]{images/MOSFETs.jpg}\\ \vspace{\baselineskip} \footnotesize\em \myhref{https://commons.wikimedia.org/wiki/File:MOSFETs.jpg}{Dick Lyon} [Public domain] \end{center} } \hfill \mbox{} \end{center} } \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Lauko ir bipoliariniai tranzistoriai} \begin{center} \hfill \includegraphics[height=4cm]{drawings/transistors/transistor-marked-termini_lt.eps} \hfill \includegraphics[height=4cm]{drawings/transistors/bipolar-n-p-n_lt.eps} \hfill \includegraphics[height=4cm]{drawings/transistors/bipolar-p-n-p_lt.eps} \hfill \mbox{} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Tranzistoriaus savybės} \only<1>{ \begin{center} \hfill \includegraphics[height=5cm]{drawings/transistors/transistor-closed.eps} \hfill \includegraphics[height=5cm]{drawings/mechanical-switches/switch.eps} \hfill \mbox{} \end{center} } \only<2>{ \begin{center} \hfill \includegraphics[height=5cm]{drawings/transistors/transistor-open.eps} \hfill \includegraphics[height=5cm]{drawings/mechanical-switches/switch-closed.eps} \hfill \mbox{} \end{center} } \only<3>{ \begin{center} \hfill \includegraphics[height=5cm]{drawings/transistors/transistor-open-with-current.eps} \hfill \includegraphics[height=5cm]{drawings/mechanical-switches/switch-closed-with-current.eps} \hfill \mbox{} \end{center} } \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Du tranzistorių tipai} \begin{center} \begin{tabular}{cccccc} \parbox[c]{2cm}{ \begin{center} \includegraphics[height=5cm]{drawings/transistors/transistor.eps} \end{center} } & {\Large$\Leftrightarrow$} & \parbox[c]{2cm}{ \begin{center} \includegraphics[height=5cm]{drawings/mechanical-switches/switch.eps} \end{center} } & \parbox[c]{2cm}{ \begin{center} \includegraphics[height=5cm]{drawings/transistors/transistor-pmos.eps} \end{center} } & {\Large$\Leftrightarrow$} & \parbox[c]{2cm}{ \begin{center} \includegraphics[height=5cm]{drawings/mechanical-switches/switch-normally-closed.eps} \end{center} } \\ \multicolumn{2}{c}{ n-tipo tranzistorius } && \multicolumn{2}{c}{ p-tipo tranzistorius } & \\ \end{tabular} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{N-tipo ir p-tipo tranzistorių palyginimas} \only<1>{ \begin{center} \includegraphics[height=5cm]{drawings/transistors/transistors-n-type-p-type-open-with-current.eps} \end{center} } \only<2>{ \begin{center} \includegraphics[height=5cm]{drawings/transistors/transistors-n-type-p-type-closed.eps} \end{center} } \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{N-tipo ir p-tipo tranzistorių apžvalga} \begin{center} \hfill \includegraphics[height=5cm]{drawings/transistors/transistors-n-type-p-type-open-with-current.eps} \hfill \includegraphics[height=5cm]{drawings/transistors/transistors-n-type-p-type-closed.eps} \hfill \mbox{} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Tranzistorinių perjungiklių įtampos} \only<1>{ \begin{center} \hfill \includegraphics[height=5cm]{drawings/transistors/transistor-switch-voltages_lt.eps} \hfill \includegraphics[height=5cm]{drawings/mechanical-switches/mechanical-switch-voltages_lt.eps} \hfill \mbox{} \end{center} } \only<2>{ \begin{center} \hfill \includegraphics[height=5cm]{drawings/transistors/transistor-switch-voltages-off_lt.eps} \hfill \includegraphics[height=5cm]{drawings/mechanical-switches/mechanical-switch-voltages-off_lt.eps} \hfill \mbox{} \end{center} } \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Priešingos tranzistorinio perjungiklio įtampos} \begin{center} \hfill \includegraphics[height=5cm]{drawings/transistors/transistor-switch-voltages-off-coloured_lt.eps} \hfill \includegraphics[height=5cm]{drawings/transistors/transistor-switch-voltages-coloured_lt.eps} \hfill \mbox{} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Invertorius (loginė f-ja NE/NOT)} \begin{center} \hfill \begin{tabular}{ll} L: & Žemės potencialas ($\approx$ 0V) \\ H: & Maitinimo įtampa (5V, 3.3V, ...) \\ \end{tabular} \hfill \begin{tabular}{ll} L: & 0 \\ H: & 1 \\ \end{tabular} \hfill \begin{tabular}{ll} L: & False \\ H: & True \\ \end{tabular} \hfill \mbox{} \end{center} \begin{center} \hfill \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline L & H \\ H & L \\ \end{tabular} \hfill \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline 0 & 1 \\ 1 & 0 \\ \end{tabular} \hfill \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline False & True \\ True & False \\ \end{tabular} \hfill \mbox{} \end{center} \begin{center} \hfill \includegraphics[height=3.5cm]{drawings/transistors/transistor-switch-voltages-off-coloured_lt.eps} \hfill \includegraphics[height=3.5cm]{drawings/transistors/transistor-switch-voltages-coloured_lt.eps} \hfill \mbox{} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} %%BEGIN LANGUAGE lt \frametitle{Istorinės loginės schemos: Apolono valdymo kompiuteris} %%END LANGUAGE lt \begin{center} \begin{tabular}{cc} \includegraphics[height=3cm]{images/Agc_nor2a_half.jpg} & \rule{0pt}{3.1cm} \hspace{0.8cm} \includegraphics[height=3cm]{images/Agc_nor2.jpg} \hspace{0.8cm} \\ \includegraphics[height=3cm]{images/Agc_flatp.jpg} & \only<1>{ \rule{0pt}{3.1cm} \includegraphics[height=3cm]{images/Agc_view.jpg} } \only<2>{ \rule{0pt}{3.1cm} \includegraphics[height=3cm]{images/596px-Apollo_11_Lunar_Lander_-_5927_NASA_gimp.jpeg} } \end{tabular} \end{center} \scriptsize \only<1>{ All AGC images: Wikipedia. \myhref{https://en.wikipedia.org/wiki/Apollo\_Guidance\_Computer}{Apollo guidance computer}, (viewed 2020-08-10). All images by NASA, public domain. } \only<2>{ Lunar Landing Module image: by \myhref{https://commons.wikimedia.org/wiki/File:Apollo_11_Lunar_Lander_-_5927_NASA.jpg}{Neil Armstrong} \newline (viewed 2020-08-10). [Public domain] } \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Išsklaidyta galia} \begin{center} \hfill \includegraphics[height=5cm]{drawings/transistors/transistor-switch-voltages-heat_lt.eps} \hfill \includegraphics[height=5cm]{drawings/transistors/transistor-totem-pole-switch.eps} \hfill \mbox{} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{CMOS ventilis NE/NOT} \begin{center} \hfill \includegraphics[height=5cm]{drawings/transistors/CMOS-NOT-gate.eps} \hfill \includegraphics[height=5cm]{drawings/transistors/CMOS-NOT-gate-input-low.eps} \hfill \mbox{} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{NOT ventilis, sumodeliuotas Logisim} \myhref{http://www.cburch.com/logisim/}{Logisim} %%BEGIN LANGUAGE lt yra puiki programa skaitmeninių įrenginių konstravimui ir modeliavimui. %%END LANGUAGE lt \begin{center} \includegraphics[width=5cm]{drawings/circuits/CMOS-NOT-gate-input-High.png} \hspace{0.5em} \includegraphics[width=5cm]{drawings/circuits/CMOS-NOT-gate-input-Low.png} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Loginis elementas NE – priminimas} \begin{center} \hfill \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline L & H \\ H & L \\ \end{tabular} \hfill \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline 0 & 1 \\ 1 & 0 \\ \end{tabular} \hfill \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline False & True \\ True & False \\ \end{tabular} \hfill \mbox{} \end{center} \only<1>{ \begin{center} \hfill \includegraphics[height=4.5cm]{drawings/transistors/CMOS-NOT-gate-input-low.eps} \hfill \includegraphics[height=4.5cm]{drawings/transistors/CMOS-NOT-gate.eps} \hfill \mbox{} \end{center} } \only<2>{ \begin{center} \hfill \begin{minipage}[c]{0pt} \rule[-0.5cm]{0pt}{4.5cm} \end{minipage} \begin{minipage}[c]{3.5cm} \includegraphics[width=\textwidth]{drawings/circuits/CMOS-NOT-gate-input-Low.png} \end{minipage} \hspace{0.5em} \begin{minipage}[c]{3.5cm} \includegraphics[width=\textwidth]{drawings/circuits/CMOS-NOT-gate-input-High.png} \end{minipage} \hfill \mbox{} \end{center} } \only<3>{ \begin{center} \hfill \begin{minipage}[c]{0pt} \rule[-0.5cm]{0pt}{4.5cm} \end{minipage} \begin{minipage}[c]{3.5cm} \includegraphics[width=\textwidth]{drawings/circuits/CMOS-NOT-gate-input-High.png} \end{minipage} \hspace{0.5em} \begin{minipage}[c]{3.5cm} \includegraphics[width=\textwidth]{images/gates/NOT-gate-both-states-Logisim.png} \end{minipage} \hfill \mbox{} \end{center} } \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Būlio funkcijos} %%BEGIN LANGUAGE lt Paremtos dviejų elementų Būlio algebra, arba „perjungiklių/relių algebra“. %%END LANGUAGE lt \vspace{-2\baselineskip} \begin{center} $$ \begin{array}{c} B = \left\{ 0, 1 \right\} \\ B \times \dots \times B \mapsto B \\ \end{array} $$ \end{center} \hfill \begin{minipage}[t]{0.25\textwidth} \mbox{} \includegraphics[width=\linewidth]{images/De_Morgan_Augustus.jpg} \scriptsize \myhref{https://en.wikipedia.org/wiki/Augustus_De_Morgan}{Augustus De Morgan} \end{minipage} \hfill \begin{minipage}[t]{0.25\textwidth} \mbox{} \includegraphics[width=\linewidth]{images/ClaudeShannon_MFO3807.jpg} \scriptsize \myhref{https://en.wikipedia.org/wiki/Claude_Shannon}{Claude Shannon} \tiny \vspace{0.5\baselineskip} \rightline{ \parbox{0.83\linewidth}{ \raggedright\em Photo by Konrad Jacobs, reused \myhref{https://commons.wikimedia.org/wiki/File:ClaudeShannon_MFO3807.jpg}{with permission} } } \end{minipage} \hfill \begin{minipage}[t]{0.25\textwidth} \mbox{} \includegraphics[width=\linewidth]{images/George_Boole_color.jpg} \scriptsize \myhref{https://en.wikipedia.org/wiki/George_Boole}{George Boole} \end{minipage} \hfill \mbox{} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Vieno argumento funkcijų skaičius} %%\tiny %%\small \vspace{-1\baselineskip} \begin{center} % "When matrix is used in a frame environment, you need ampersand % replacement just as you would when the matrix is in the % argument to a command" % (https://tex.stackexchange.com/questions/208408/tikz-matrix-undefined-control-sequence): \begin{tikzpicture}[column sep=-2em, ampersand replacement=\&] \matrix { \node(table) { \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline 0 & 1 \\ 1 & 0 \\ \end{tabular} }; \& \node(formula) { \parbox{17em}{ \vspace{\baselineskip} $$ \left\} \begin{array}{l} \\ \\ \end{array} \right. \hspace{-1em} 2 \,\,\text{positions} \Rightarrow 2^2 = 4 \,\,\text{combinations} $$ } }; \\ }; \end{tikzpicture} %%\tiny \footnotesize \newcommand{\keeprowheight}{\rule{0pt}{2.5em}} \newlength{\logicbaselength} \setlength{\logicbaselength}{2.5cm} %\vspace{-0.5\baselineskip} \begin{tabular}{llll} \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline 0 & 0 \\ 1 & 1 \\ \end{tabular} & identity & $f(x\,)=x$ & \parbox[c]{\logicbaselength}{ \includegraphics[width=\linewidth]{drawings/logic-gates/identity-Boolean-function.png} } \\ \keeprowheight \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline 0 & 1 \\ 1 & 0 \\ \end{tabular} & NOT & $f(x\,)=\overline{x\,}$ & \parbox[c]{\logicbaselength}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NOT-Boolean-function.png} } \\ \keeprowheight \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline 0 & 0 \\ 1 & 0 \\ \end{tabular} & constant 0 & $f(x\,)=0$ & \parbox[c]{\logicbaselength}{ \includegraphics[width=\linewidth]{drawings/logic-gates/constant-0-Boolean-function.png} } \\ \keeprowheight \begin{tabular}{c|c} \textbf{Įvestis} & \textbf{Išvestis} \\ \hline 0 & 1 \\ 1 & 1 \\ \end{tabular} & constant 1 & $f(x\,)=1$ & \parbox[c]{\logicbaselength}{ \includegraphics[width=\linewidth]{drawings/logic-gates/constant-1-Boolean-function.png} } \\ \end{tabular} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Loginė funkcija IR-NE} \begin{center} \footnotesize $X = f(A,B) = \overline{A \cdot B} = \overline{AB}$ \vspace{0.5\baselineskip} \begin{tabular}{c|c|c} \textbf{A} & \textbf{B} & \textbf{X}\\ \hline \only<6>{\bf} 0 & \only<6>{\bf} 0 & \only<6>{\bf} 1 \\ \only<7>{\bf} 0 & \only<7>{\bf} 1 & \only<7>{\bf} 1 \\ \only<8>{\bf} 1 & \only<8>{\bf} 0 & \only<8>{\bf} 1 \\ \only<9>{\bf} 1 & \only<9>{\bf} 1 & \only<9>{\bf} 0 \\ \end{tabular} \vspace{\baselineskip} \newcommand{\keeprowheight}{\rule[-0.2cm]{0pt}{4cm}} \only<1>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NAND-gate-half-A1-B1.png} } \only<2>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NAND-gate-half-A1-B0.png} } \only<3>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NAND-gate-half-A0-B1.png} } \only<4>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NAND-gate-half-A0-B0.png} } \only<5,9>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NAND-gate-A1-B1.png} } \only<6>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NAND-gate-A0-B0.png} } \only<7>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NAND-gate-A0-B1.png} } \only<8>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NAND-gate-A1-B0.png} } \only<10>{ \parbox[c]{4cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NAND-gate-A0-B0.png} } \hspace{2em} \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NAND-gate-A0-B0.png} } } \only<11>{ \parbox[c]{4cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NAND-gate-A0-B0.png} } \hspace{2em} \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NAND-gate-A0-B0.png} \includegraphics[width=\linewidth]{drawings/logic-gates/NAND-gate-A0-B1.png} \includegraphics[width=\linewidth]{drawings/logic-gates/NAND-gate-A1-B1.png} } } \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Loginė funkcija ARBA-NE} \begin{center} \footnotesize $X = f(A,B) = \overline{A \lor B}$ \vspace{0.5\baselineskip} \begin{tabular}{c|c|c} \textbf{A} & \textbf{B} & \textbf{X}\\ \hline \only<2>{\bf} 0 & \only<2>{\bf} 0 & \only<2>{\bf} 1 \\ \only<3>{\bf} 0 & \only<3>{\bf} 1 & \only<3>{\bf} 0 \\ \only<4>{\bf} 1 & \only<4>{\bf} 0 & \only<4>{\bf} 0 \\ \only<5>{\bf} 1 & \only<5>{\bf} 1 & \only<5>{\bf} 0 \\ \end{tabular} \vspace{\baselineskip} \newcommand{\keeprowheight}{\rule[-0.2cm]{0pt}{4cm}} \only<1-2>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NOR-gate-A0-B0.png} } \only<3>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NOR-gate-A0-B1.png} } \only<4>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NOR-gate-A1-B0.png} } \only<5>{ \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-NOR-gate-A1-B1.png} } \only<6>{ \parbox[c]{4cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NOR-gate-A0-B0.png} } \hspace{2em} \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NOR-gate-A0-B0.png} } } \only<7>{ \parbox[c]{4cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NOR-gate-A0-B0.png} } \hspace{2em} \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NOR-gate-A0-B0.png} \includegraphics[width=\linewidth]{drawings/logic-gates/NOR-gate-A0-B1.png} \includegraphics[width=\linewidth]{drawings/logic-gates/NOR-gate-A1-B1.png} } } \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Loginė funkcija ARBA (disjunkcija)} \begin{center} \footnotesize $X = f(A,B) = A \lor B$ \vspace{0.5\baselineskip} \begin{tabular}{c|c|c} \textbf{A} & \textbf{B} & \textbf{X}\\ \hline \only<4>{\bf} 0 & \only<4>{\bf} 0 & \only<4>{\bf} 0 \\ \only<5>{\bf} 0 & \only<5>{\bf} 1 & \only<5>{\bf} 1 \\ \only<6>{\bf} 1 & \only<6>{\bf} 0 & \only<6>{\bf} 1 \\ \only<7>{\bf} 1 & \only<7>{\bf} 1 & \only<7>{\bf} 1 \\ \end{tabular} \vspace{\baselineskip} \newcommand{\keeprowheight}{\rule[-0.2cm]{0pt}{4cm}} \begin{tikzpicture}[column sep=-1em, ampersand replacement=\&] \only<1>{ \node(image) { \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-OR-gate-A0-B0.png} }; } \only<2>{ \node(image) { \keeprowheight \includegraphics[width=4cm]{drawings/circuits/CMOS-OR-gate-A0-B0.png} }; \draw [red,thick] (2,2) to (-2,-2); \draw [red,thick] (2,-2) to (-2,2); } \end{tikzpicture} \only<3-4>{ \keeprowheight \includegraphics[width=5cm]{drawings/circuits/CMOS-NOR-NOT-gates-A0-B0.png} } \only<5>{ \keeprowheight \includegraphics[width=5cm]{drawings/circuits/CMOS-NOR-NOT-gates-A0-B1.png} } \only<6>{ \keeprowheight \includegraphics[width=5cm]{drawings/circuits/CMOS-NOR-NOT-gates-A1-B0.png} } \only<7>{ \keeprowheight \includegraphics[width=5cm]{drawings/circuits/CMOS-NOR-NOT-gates-A1-B1.png} } \only<8>{ \parbox[c]{5cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NOR-NOT-gates-A0-B0.png} } \hspace{2em} \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/OR-gate-A0-B0.png} } } \only<9>{ \parbox[c]{5cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NOR-NOT-gates-A0-B0.png} } \hspace{2em} \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/OR-gate-A0-B0.png} \includegraphics[width=\linewidth]{drawings/logic-gates/OR-gate-A0-B1.png} \includegraphics[width=\linewidth]{drawings/logic-gates/OR-gate-A1-B1.png} } } \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Dviejų argumentų funkcijos} %%\tiny %%\small \vspace{-1\baselineskip} \begin{center} % "When matrix is used in a frame environment, you need ampersand % replacement just as you would when the matrix is in the % argument to a command" % (https://tex.stackexchange.com/questions/208408/tikz-matrix-undefined-control-sequence): \begin{tikzpicture}[column sep=-1em, ampersand replacement=\&] \matrix { \node(table) { \begin{tabular}{c|c|c} \textbf{A} & \textbf{B} & \textbf{X} \\ \hline 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{tabular} }; \& \node(formula) { \parbox{17em}{ \vspace{\baselineskip} $$ \left\} \begin{array}{l} \\ \\ \\ \\ \end{array} \right. \hspace{-1em} 4 \,\,\text{positions} \Rightarrow 2^{2^2} = 16 \,\,\text{combinations} $$ } }; \\ }; \end{tikzpicture} \parbox[c]{5cm}{ \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NOR-NOT-gates-A0-B0.png} } \parbox[c]{5cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NOR-NOT-A0-B0.png} } \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Daugelio įėjimų loginės grandinės} \vspace{-1\baselineskip} \begin{center} \begin{tabular}{cccc|c} \textbf{A} & \textbf{B} & \textbf{C} & \textbf{D} & \textbf{X} \\ \hline 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \multicolumn{4}{c}{\dots} \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ \end{tabular} \vfill \hfill \parbox[c]{1em}{ \renewcommand{\arraystretch}{1.1} \begin{tabular}{r} \textbf{A} \\ \textbf{B} \\ \textbf{C} \\ \textbf{D} \\ \end{tabular} } \parbox[c]{0.3\textwidth}{ \includegraphics[width=\linewidth]{drawings/logic-gates/4-input-AND-from-2input-gates-one-0.png} } \hfill \parbox[c]{0.3\textwidth}{ \includegraphics[width=\linewidth]{drawings/logic-gates/4-input-AND-from-2input-gates-all-1.png} } \hfill \mbox{} \vfill \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Grandinė bet kokiai loginei funkcijai apskaičiuoti} \newcommand{\termOne}{\overline{x_1}\cdot\overline{x_2}\cdot ... \cdot\overline{x_n}} \newcommand{\termTwo}{\overline{x_1}\cdot x_2 \cdot ... \cdot\overline{x_n}} \newcommand{\termN}{x_1\cdot x_2 \cdot ... \cdot\overline{x_n}} \vspace{-1\baselineskip} \begin{center} \begin{tabular}{cccc|cc} \boldm{$x_1$} & \boldm{$x_2$} & \textbf{\dots} & \boldm{$x_n$} & \boldm{$f(x_1,x_2,\dots,x_n)$} & \textbf{konjunkcija} \\ \hline 0 & 0 & \dots & 0 & 1 & $\termOne$ \\ 0 & 1 & \dots & 0 & 1 & $\termTwo$ \\ 0 & 1 & \dots & 0 & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 & 0 \\ \multicolumn{4}{c|}{\dots} & \multicolumn{2}{c}{\dots} \\ 1 & 1 & \dots & 0 & 1 & $\termN$ \\ 1 & 1 & \dots & 1 & 0 & 0 \\ \end{tabular} \vfill \vfill Disjunkcinė normalioji forma (DNF): $$ \begin{array}{rl} f( x_1, x_2, \dots, x_n ) = & \termOne \\ \lor & \termTwo \\ \lor & \dots \\ \lor & \termN \end{array} $$ \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Pavyzdys: funkcijos XOR (griežtosios disjunkcijos) sintezė} \begin{center} \begin{tabular}{cc|cc} \textbf{A} & \textbf{B} & \textbf{X} & \textbf{konjunkcija} \\ \hline 0 & 0 & 0 & \\ 0 & 1 & 1 & $\overline{A} \cdot B$ \\ 1 & 0 & 1 & $A \cdot \overline{B}$ \\ 1 & 1 & 0 & \\ \end{tabular} \vspace{\baselineskip} DNF: $ X = \overline{A} B \lor A \overline{B} $ \vspace{\baselineskip} \only<2->{ \parbox[c]{1em}{ \footnotesize \renewcommand{\arraystretch}{1.7} \begin{tabular}{r} \textbf{A} \\ \textbf{B} \\ \\ \\ \end{tabular} } } \begin{minipage}{0.5\textwidth} \only<2,6->{ \parbox[c]{\textwidth}{ \includegraphics[width=\linewidth]{drawings/logic-gates/XOR-synthesis-A0-B0.png} } } \only<3>{ \parbox[c]{\textwidth}{ \includegraphics[width=\linewidth]{drawings/logic-gates/XOR-synthesis-A0-B1.png} } } \only<4>{ \parbox[c]{\textwidth}{ \includegraphics[width=\linewidth]{drawings/logic-gates/XOR-synthesis-A1-B0.png} } } \only<5>{ \parbox[c]{\textwidth}{ \includegraphics[width=\linewidth]{drawings/logic-gates/XOR-synthesis-A1-B1.png} } } \end{minipage} \only<6>{ \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/XOR-A0-B0.png} } } \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Loginis ventilis IR-NE} \newcommand{\keeprowheight}{\rule[-0.2cm]{0pt}{3.7cm}} \begin{center} $X = f(A,B) = \overline{AB}$ \vspace{\baselineskip} \begin{tabular}{c|c|c} \textbf{A} & \textbf{B} & \textbf{X} \\ \hline \only<2>{\bf} 0 & \only<2>{\bf} 0 & \only<2>{\bf} 1 \\ \only<3>{\bf} 0 & \only<3>{\bf} 1 & \only<3>{\bf} 1 \\ \only<4>{\bf} 1 & \only<4>{\bf} 0 & \only<4>{\bf} 1 \\ \only<5>{\bf} 1 & \only<5>{\bf} 1 & \only<5>{\bf} 0 \\ \end{tabular} \vspace{\baselineskip} \only<1>{ \parbox[c]{5cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-AND-NOT-gates-A0-B0.png} } \parbox[c]{5cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/AND-NOT-A0-B0.png} } } \only<2>{ \parbox[c]{4cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NAND-gate-A0-B0.png} } } \only<2>{ \parbox[c]{4cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NAND-gate-A0-B0.png} } } \only<3>{ \parbox[c]{4cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NAND-gate-A0-B1.png} } } \only<3>{ \parbox[c]{4cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NAND-gate-A0-B1.png} } } \only<4>{ \parbox[c]{4cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NAND-gate-A1-B0.png} } } \only<4>{ \parbox[c]{4cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NAND-gate-A1-B0.png} } } \only<5>{ \parbox[c]{4cm}{ \keeprowheight \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NAND-gate-A1-B1.png} } } \only<5>{ \parbox[c]{4cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NAND-gate-A1-B1.png} } } \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{NAND – universalus loginis ventilis} \begin{center} \begin{tabular}{c|c|c} \textbf{A} & \textbf{B} & \textbf{X} \\ \hline 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{tabular} \vfill \begin{tabular}{lcl} \parbox[c]{4cm}{ \includegraphics[height=1cm]{drawings/logic-gates/NOT-from-NAND-input-0.png} } & $\Leftrightarrow$ & \parbox[c]{3cm}{ \includegraphics[width=0.7\linewidth]{drawings/logic-gates/NOT-Boolean-function.png} } \\ \parbox[c]{4cm}{ \includegraphics[height=1cm]{drawings/logic-gates/AND-from-NAND-A0-B0.png} } & $\Leftrightarrow$ & \parbox[c]{3cm}{ \includegraphics[height=1cm]{drawings/logic-gates/AND-gate-A0-B0.png} } \\ \parbox[c]{4cm}{ \includegraphics[height=1.5cm]{drawings/logic-gates/OR-from-NAND-A0-B0.png} } & $\Leftrightarrow$ & \parbox[c]{3cm}{ \includegraphics[height=1cm]{drawings/logic-gates/OR-gate-A0-B0.png} } \end{tabular} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Dualūs ventiliai} \begin{center} \begin{tabular}{cc|cc|c|c|c|c|c} \boldm{$A$} & \boldm{$B$} & \boldm{$\overline{A}$} & \boldm{$\overline{B}$} & \boldm{$\overline{A \cdot B}$} & \boldm{$\overline{A}\cdot\overline{B}$} & \boldm{$\overline{\overline{A}\cdot\overline{B}}$} & \boldm{$A \lor B$}\\ \hline 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ \end{tabular} \end{center} \begin{center} \parbox[c]{0pt}{ \rule{0pt}{3.5cm} } \begin{minipage}[c]{0.7\textwidth} \begin{center} \only<1>{ \parbox[c]{4cm}{ \rightline{ \includegraphics[height=1.5cm]{drawings/logic-gates/OR-from-NAND-A0-B0.png} } } \parbox[c]{3cm}{ \includegraphics[height=1cm]{drawings/logic-gates/OR-gate-A0-B0.png} } } \only<2>{ \parbox[c]{4cm}{ \rightline{ \includegraphics[height=1cm]{drawings/logic-gates/NOT-AND-NOT-A0-B0.png} } } \parbox[c]{3cm}{ \includegraphics[height=1cm]{drawings/logic-gates/OR-gate-A0-B0.png} } } \only<3>{ \parbox[c]{4cm}{ \rightline{ \includegraphics[height=1cm]{drawings/logic-gates/NOT-AND-NOT-A0-B1.png} } } \parbox[c]{3cm}{ \includegraphics[height=1cm]{drawings/logic-gates/OR-gate-A0-B1.png} } } \only<4>{ \parbox[c]{4cm}{ \rightline{ \includegraphics[height=1cm]{drawings/logic-gates/NOT-AND-NOT-A1-B0.png} } } \parbox[c]{3cm}{ \includegraphics[height=1cm]{drawings/logic-gates/OR-gate-A1-B0.png} } } \only<5>{ \parbox[c]{4cm}{ \rightline{ \includegraphics[height=1cm]{drawings/logic-gates/NOT-AND-NOT-A1-B1.png} } } \parbox[c]{3cm}{ \includegraphics[height=1cm]{drawings/logic-gates/OR-gate-A1-B1.png} } } \only<6>{ \parbox[c]{4cm}{ \rightline{ \includegraphics[height=1cm]{drawings/logic-gates/NOT-AND-NOT-A0-B0.png} } } \parbox[c]{3cm}{ \includegraphics[height=1cm]{drawings/logic-gates/OR-gate-A0-B0.png} } } \only<6>{ $$ \begin{array}{rclcl} A \lor B &=& \overline{\overline{A} \cdot \overline{B}} \\ \overline{\overline{A}} &=& A \\ \overline{A \lor B} &=& \overline{\overline{\overline{A} \cdot \overline{B}}} &=& \overline{A} \cdot \overline{B} \end{array} $$ } \end{center} \end{minipage} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Dualūs loginiai elementai, De Morgano dėsniai} \newcommand{\ol}{\overline} \begin{center} \begin{tabular}{ccc} \parbox[c]{2.5cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NOT-NAND-A0-B0.png} } & \parbox[c]{1em}{ $\Leftrightarrow$ } & \parbox[c]{2.5cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/OR-gate-A0-B0.png} } \\ \\ \parbox[c]{2.5cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NOT-NOR-A0-B0.png} } & \parbox[c]{1em}{ $\Leftrightarrow$ } & \parbox[c]{2.5cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/AND-gate-A0-B0.png} } \end{tabular} \vspace{0.5\baselineskip} $$ \left. \begin{array}{ccc} \ol{\ol{A} \cdot \ol{B}} &=& A \lor B \\ \ol{\ol{A} \lor \ol{B}} &=& A \cdot B \end{array} \right\} \,\text{ dualios funkcijos } $$ \vspace{-0.5\baselineskip} $$ \left. \begin{array}{ccc} \ol{A \lor B} &=& \ol{A} \cdot \ol{B} \\ \ol{A \cdot B} &=& \ol{A} \lor \ol{B} \end{array} \right\} \,\text{ De Morgano dėsniai } $$ \vspace{-\baselineskip} $$ h \ \ \text{ yra duali }\ \ f \ \ \underset{\text{def}}{\Leftrightarrow}\ % h(x_1,x_2,\dots,x_n) = \ol{f(\ol{x_1},\ol{x_2},\dots,\ol{x_n})} $$ \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Dar vienas universalus elementas: NOR} \begin{center} \begin{tabular}{ccc} \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NAND-gate-A0-B0.png} } & duali & \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NOR-gate-A0-B0.png} } \\ \\ \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/AND-gate-A0-B0.png} } & duali & \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/OR-gate-A0-B0.png} } \\ \\ \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NOT-Boolean-function.png} } & duali & \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/NOT-Boolean-function.png} } \end{tabular} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Pilna loginių funkcijų sistemos} Superpozicija: $$h(x_1,\dots,x_n) = f(g_1(x_1,\dots,x_n),\dots,g_m(x_1,\dots,x_n))$$ Pavyzdys: $$ \text{NOT}(\text{AND}(x_1,\text{NOT}(x_2)) \,\underset{\text{def}}{\equiv}\, \overline{x_1\overline{x_2}} = \overline{x_1}\lor\overline{\overline{x_2}} = \overline{x_1} \lor x_2 $$ Pilnos funkcijų sistemos (pavyzdžiai): \begin{center} \small \begin{tabular}{cl} $\left\{ \text{AND}, \text{OR}, \text{NOT} \right\} = \left\{ \cdot, \lor, \overline{X\,} \right\}$ & (neminimali!) \\ $\left\{ \text{AND}, \text{NOT} \right\} = \left\{ \cdot, \overline{X\,} \right\}$ \\ $\left\{ \text{OR}, \text{NOT} \right\} = \left\{ \lor, \overline{X\,} \right\}$ \\ $\left\{ \text{AND}, \text{XOR}, 1 \right\} = \left\{ \cdot, \oplus, 1 \right\}$ \\ $\left\{ \text{OR}, \text{XOR}, 1 \right\} = \left\{ \lor, \oplus, 1 \right\}$ \\ $\left\{ \text{NAND} \right\} = \left\{ \uparrow \right\}$ & \myhref{https://en.wikipedia.org/wiki/Sheffer_stroke}{Sheffer stroke} \\ $\left\{ \text{NOR} \right\} = \left\{ \downarrow \right\}$ & \myhref{https://en.wikipedia.org/wiki/Logical_NOR}{Peirce's arrow/Quine's dagger} \end{tabular} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Uždaros Būlio funkcijų klasės} %%\footnotesize \small Funkcijų klasės, uždaros superpozicijos atžvilgiu: \newcommand{\ol}{\overline} \begin{enumerate} \item $T_0$: išsaugančios nulį: $$ f(0,0,\dots, 0) = 0 $$ \item $T_1$: išsaugančios vienetą: $$ f(1,1,\dots, 1) = 1 $$ \item $S$: dualios sau: $$ f(x_1,x_2,\dots, x_n) = \ol{f(\ol{x_1},\ol{x_2},\dots,\ol{x_n})} $$ \item $L$: tiesinės: $$ f(x_1,x_2,\dots, x_n) = a_0 \oplus a_1 x_1 \oplus \dots \oplus a_n x_n $$ \item $M$: monotoninės: $$ \begin{array}{c} \vec{a} \underset{\text{def}}{\equiv} (a_1, a_2, \dots, a_n ) \\ \vec{a} \leq \vec{b} \underset{\text{def}}{\Leftrightarrow} \forall i \in \{1,\dots,n\}: a_i \leq b_i \\ f \in M \underset{\text{def}}{\Leftrightarrow} \vec{a} \leq \vec{b} \Rightarrow f(\vec{a}) \leq f(\vec{b}) \end{array} $$ \end{enumerate} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Posto teorema} \small %%BEGIN LANGUAGE lt Būlio funkcijų aibė $B = \{ f_1, f_2, \dots, f_n \}$ yra pilna superpozicijos atžvilgiu \textbf{tada ir tik tada}, jeigu: %%END LANGUAGE lt \begin{enumerate} \item joje yra funkcija, neišsauganti nulio: $$ \exists\, f_i \in B: f_i \not\in T_0 $$ \item joje yra funkcija, neišsauganti vieneto: $$ \exists\, f_i \in B: f_i \not\in T_1 $$ \item joje yra funkcija, neduali sau pačiai: $$ \exists\, f_i \in B: f_i \not\in S $$ \item joje yra netiesinė funkcija: $$ \exists\, f_i \in B: f_i \not\in L $$ \item joje yra nemonotoninė funkcija: $$ \exists\, f_i \in B: f_i \not\in M $$ \end{enumerate} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{``Apolono'' valdymo kompiuterio ventiliai} \begin{center} \includegraphics[height=3cm]{images/Agc_nor2a.jpg} \vspace{0.5\baselineskip} \parbox[c]{2.5cm}{ \includegraphics[width=\linewidth]{images/Agc_flatp.jpg} } \hspace{2em} \parbox[c]{7cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/3-input-NOR-RTL-gate-all-0.png} } \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Ką mes iki šiol pasiekėme ...} \begin{center} \begin{tabular}{cc} \parbox[c]{3cm}{ \includegraphics[width=\linewidth]{drawings/transistors/transistor-switch-voltages-coloured.eps} } & \parbox[c]{4cm}{ \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NOT-gate-input-Low.png} } \\ \parbox[c]{4cm}{ \includegraphics[width=\linewidth]{drawings/circuits/CMOS-NAND-gate-A0-B1.png} } & \parbox[c]{4cm}{ \includegraphics[width=\linewidth]{drawings/logic-gates/XOR-synthesis-A1-B0.png} } \\ \end{tabular} \end{center} \end{frame} %------------------------------------------------------------------------------ \begin{frame} \frametitle{Apibendrinimas} \begin{itemize} \item %%BEGIN LANGUAGE lt Skaitmenines grandines galima aprašyti Būlio algebros pagalba; %%END LANGUAGE lt \item %%BEGIN LANGUAGE lt Bet kokia Būlio funkcija gali būti sukonstruota iš pilnos funkcijų aibės (žr. Posto teoremą); loginės funkcijos NAND ir NOR kiekviena pati sudaro pilną funkcijų aibę ir leidžia sukonstruoti bet kokią loginę grandinę; %%END LANGUAGE lt \item %%BEGIN LANGUAGE lt Visas logines funkcijas gali realizuoti elektriniai perjungikliai (pvz. KMOP lauko tranzistoriai); %%END LANGUAGE lt \item %%BEGIN LANGUAGE lt Visus kompiuterio mazgus galima pagaminti iš valdomų perjungiklių; šiuolaikiniuose kompiuteriuose kaip perjungikliai naudojami KMOP lauko tranzistoriai. %%END LANGUAGE lt \end{itemize} \end{frame} %------------------------------------------------------------------------------ %% \begin{frame}[allowframebreaks] %% %%LANGUAGE en \frametitle{References} %% %%LANGUAGE lt \frametitle{Šaltiniai} %% %%LANGUAGE ru \frametitle{Источники} %% %% %%\scriptsize %% %%\bibliographystyle{plain} %% %%\bibliographystyle{apalike} %% %%\bibliography{bibliography/citations} %% \renewcommand{\bibfont}{\scriptsize} %% \printbibliography %% %% \end{frame} %------------------------------------------------------------------------------ \end{document} % 2023-09-11 18:14:50 EEST