First version of defense

Obhajoba/slides.tex

-\documentclass{beamer}
+\documentclass[aspectratio=169]{beamer}
 %\usepackage[czech]{babel}
 \usepackage[utf8]{inputenc}
 %balíček na A4
@@ -13,7 +13,7 @@
 
 %\usepackage[mathlf]{MinionPro}
 \usepackage{sansmathfonts}
-\usepackage[mathlf,textlf]{MyriadPro}
+%\usepackage[mathlf,textlf]{MyriadPro}
 %\usepackage{mathpazo}
 \usepackage[small]{eulervm}
 %\usepackage{newtxmath}
@@ -28,6 +28,18 @@
 \usepackage{tabularx}
 \usepackage{pbox}
 
+\usepackage{tikz}
+
+\tikzset{
+   invisible/.style={opacity=0},
+   visible on/.style={alt=#1{}{invisible}},
+   blank/.style={},
+   red on/.style={alt={#1{fill=red!10}{blank}}},
+   alt/.code args={<#1>#2#3}{%
+     \alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}} % \pgfkeysalso doesn't change the path
+   },
+ }
+
 \captionsetup{compatibility=false}
 %\renewcommand{\arraystretch}{1}
 %\hypersetup{pdfpagemode={FullScreen}}
@@ -111,9 +123,261 @@
 
 \frame{\titlepage}
 
-\section{Preliminaries}
+\section{Propositional Satisfiability}
 \begin{frame}
-  Test
+  \frametitle{Propositional Satisfiability}
+
+  Propositional satisfiability problem (SAT) -- decide whether a
+  propositional formula is satisfiable.
+
+  \bigskip
+
+  For example
+  $\varphi = (y \vee z)~\wedge~(\neg y \vee z)~\wedge~(y \vee x
+  \vee z)~\wedge~(y \vee \neg x \vee \neg z)$ is satisfiable.
+
+  \bigskip
+
+  Usually solved by a \alert{conflict driven clause learning} algorithm (CDCL), which relies on:
+  \begin{itemize}
+  \item preprocessing,
+  \item variable assignment (with heuristics),
+  \item unit propagation,
+  \item conflict analysis,
+  \item clause learning,
+  \item backtracking,
+  \item restarts.
+  \end{itemize}
+
+  % \begin{example}
+  %   $\varphi = (\neg y \vee z)~\wedge~(\neg y \vee z)~\wedge~(y \vee x \vee z)~\wedge~(y \vee \neg x \vee \neg z)$.
+
+  %   \bigskip
+
+  %   \begin{tikzpicture}[every node/.style={draw}]
+  %     \node (e) at (5,4) {decide};
+  %     \node [visible on=<2->] (x) at (2,3) {decide};
+  %     \node [visible on=<3->] (xy) at (0,2) {propagate};
+  %     \node [visible on=<4->] (xyz) at (0,0.5) {conflict (learn $\neg y$)};
+  %     \node [visible on=<5->] (-y) at (9,3) {decide};
+  %     \node [visible on=<6->] (-yz) at (7,2) {propagate};
+  %     \node [visible on=<7->] (-yz-x) at (7,0.5) {sat};
+
+  %     \path[auto, every node/.style={}, above]
+  %     (e) edge [visible on=<2->] node {$x$} (x)
+  %     (x) edge [visible on=<3->] node {$y$} (xy)
+  %     (xy) edge [visible on=<4->, left] node {$z$} (xyz)
+  %     (e) edge [visible on=<5->] node {$\neg y$} (-y)
+  %     (-y) edge [visible on=<6->] node {$z$} (-yz)
+  %     (-yz) edge [visible on=<7->, left] node {$\neg x$} (-yz-x);
+
+  %   \end{tikzpicture}
+  % \end{example}
+\end{frame}
+
+\section{Satisfiability Modulo Theories}
+\begin{frame}
+  \frametitle{Satisfiability Modulo Theories}
+
+  Decide the satisfiability of a first-order formula in a given
+  theory.
+
+  \smallskip
+
+  Function symbols are given interpretations by the theory.
+
+  \bigskip
+
+  Traditionally solved by the \alert{CDCL modulo theories} -- combination of
+  CDCL and a specialized theory solver.
+
+  \begin{example}
+    Let $\varphi = (x + y = 3) \wedge (x > 1) \wedge (y = 2 \vee y = 1)$.
+
+    \begin{enumerate}[<+->]
+    \item Propositional model $\{ x + y = 3,~x > 1,~y = 2 \} $.
+    \item This model \textbf{is not} consistent with the theory of integers.
+    \item Conjoin $\neg (x + y = 3) \vee \neg (x > 1) \vee \neg (y = 2)$ to $\varphi$.
+    \item Propositional model $\{ x + y = 3,~x > 1,~y = 1 \} $.
+    \item This model \textbf{is} consistent with the theory of integers.
+    \item $\varphi$ is satisfiable.
+    \end{enumerate}
+  \end{example}
+\end{frame}
+
+\section{Theory of Bit-Vectors}
+\begin{frame}
+  \frametitle{Theory of Bit-Vectors}
+
+  Theory of bit-vectors describes bounded integers (or vectors of
+  bits) with:
+  \begin{itemize}
+  \item arithmetic operations,
+  \item bitwise operations,
+  \item comparison.
+  \end{itemize}
+
+  Often used in modelling of software.
+
+  \bigskip
+
+  Different from mathematical integers -- for example the formula
+  \[
+    \varphi = (x^{[32]} >_s 0)~\wedge~(y^{[32]} >_s 0)~\wedge~(x^{[32]} + y^{[32]} = 0)
+  \]
+
+  is \alert{satisfiable}.
+\end{frame}
+
+\section{Quantifier-free Bit-Vector Logic}
+\begin{frame}
+  \frametitle{Quantifier-free Bit-Vector Logic} Quantifier-free
+  bit-vector formulas usually solved by reduction to a propositional
+  formula (\emph{bit-blasting}) and using a SAT solver.
+
+  \smallskip
+
+  Approach taken by Z3, CVC4, Boolector, UCLID, Yices, \ldots.
+
+  \bigskip
+
+  Alternative approaches exist:
+  \begin{itemize}
+  \item model-constructing satisfiability calculus,
+  \item abstract CDCL,
+  \item stochastic local search.
+  \end{itemize}
+\end{frame}
+
+\section{Quantified Bit-Vector Logic}
+\begin{frame}
+  \frametitle{Quantified Bit-Vector Logic}
+
+  In many software verification applications, quantifiers are
+  necessary.
+
+  \bigskip
+
+  Quantifier bit-vector formulas traditionally solved by
+  \alert{quantifier instantiation}.
+  \begin{example}
+    Let $\varphi = 3 < a ~\wedge~ \forall x\,(a \not = 2 \times x)$
+    \pause
+    \begin{itemize}[<+->]
+    \item $3 < a$ is satisfiable with model $a = 4$.
+    \item $a = 4$ not a model of $\forall x \,(a \not = 2 \times x)$ (corresponding counter-example is $x = 2$).
+    \item Add instance of the quantifier for $x = 2$.
+    \item $3 < a ~\wedge~ (a \not = 2 \times 2)$ is satisfiable with model $a = 5$.
+    \item $a = 5$ is a model of $\forall x \,(a \not = 2 \times x)$.
+    \item $\varphi$ is satisfiable
+    \end{itemize}
+  \end{example}
+\end{frame}
+
+\section{Symbolic Approach to Quantified Bit-Vectors}
+\begin{frame}
+  \frametitle{Symbolic Approach to Quantified Bit-Vectors}
+  Bit-vector formulas can be represented by \alert{binary decision diagrams} (BDD).
+
+  \bigskip
+
+  Assuming that $x^{[4]} = x_3x_2x_1x_0$, formula
+  \[
+    x^{[4]} >_u 0
+  \]
+  is represented by
+
+  \begin{center}
+  \begin{tikzpicture}[every node/.style={draw, circle}]
+    \node (x0) at (2,4) {$x_0$};
+    \node (x1) at (3,3) {$x_1$};
+    \node (x2) at (4,2) {$x_2$};
+    \node (x3) at (5,1) {$x_3$};
+
+    \node [rectangle] (true) at (0,0) {true};
+    \node [rectangle] (false) at (6,0) {false};
+
+    \path[auto, every node/.style={}, above, dashed]
+    (x0) edge [above right] node {$0$} (x1)
+    (x1) edge [above right] node {$0$} (x2)
+    (x2) edge [above right] node {$0$} (x3)
+    (x3) edge [above right] node {$0$} (false);
+
+    \path[auto, every node/.style={}, above left]
+    (x0) edge [] node {$1$} (true)
+    (x1) edge [] node {$1$} (true)
+    (x2) edge [] node {$1$} (true)
+    (x3) edge [] node {$1$} (true);
+
+  \end{tikzpicture}
+\end{center}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Symbolic Approach to Quantified Bit-Vectors}
+
+  Efficient algorithm are known for operations with BDDs (conjunction,
+  disjunction, quantifiers).
+
+  \smallskip
+
+  Quantification usually reduces the size of the BDD -- useful for
+  quantified bit-vectors.
+
+  \bigskip
+
+  We have implemented the solver Q3B, which
+  \begin{itemize}
+  \item simplifies the formula,
+  \item converts the formula to the BDD,
+  \item tries to use approximations if the precise BDD computation is
+    too expensive.
+  \end{itemize}
+
+  Results show that Q3B is \alert{more efficient} than standard SMT solvers.
+\end{frame}
+
+\section{Aims of the Work}
+\begin{frame}
+  \frametitle{Aims of the Work}
+  \textbf{Major aims}
+  \begin{itemize}
+  \item Further development of the symbolic solver for quantified-bit
+    vectors.
+  \item Hybrid approach to quantified bit-vectors.
+  \end{itemize}
+
+  \bigskip
+
+  \textbf{Minor aims}
+  \begin{itemize}
+  \item Determining a computational complexity of quantified bit-vector logic with
+    bit-widths represented in binary.
+  \item Extending simplifications using unconstrained variables
+  \end{itemize}
+\end{frame}
+
+\section{Publications}
+\begin{frame}
+  \frametitle{Publications}
+  \begin{itemize}
+  \item \textsc{Jonáš}, M. and J. \textsc{Strejček}. \emph{``Solving
+      Quantified Bit-Vector Formulas Using Binary Decision Diagrams.''}
+    In: SAT 2016, Bordeaux, France, July 5-8, 2016, Proceedings. 2016,
+    pp. 267-283 \bigskip
+
+  Main author of the text of the paper, implemented the SMT solver and conducted all the experiments.
+\item \textsc{Chalupa} M., M. \textsc{Jonáš}, J. \textsc{Slabý},
+  J. \textsc{Strejček}, and M. \textsc{Vitovská}. \emph{``Symbiotic 3:
+    New Slicer and Error-Witness Generation -- (Competition
+    Contribution).''} In: TACAS 2016, Eindhoven, The Netherlands, April
+  2-8, 2016, Proceedings, pp. 946-949
+
+  \bigskip
+
+  Wrote parts of the paper and prepared the environment that was used
+  to run experiments with the implemented tool Symbiotic.
+\end{itemize}
 \end{frame}
 
 \end{document}