First version of defense

\documentclass{beamer}

\documentclass[aspectratio=169]{beamer}

%\usepackage[czech]{babel}

%\usepackage[czech]{babel}

\usepackage[utf8]{inputenc}

\usepackage[utf8]{inputenc}

%balíček na A4

%balíček na A4

%\usepackage[mathlf]{MinionPro}

%\usepackage[mathlf]{MinionPro}

\usepackage{sansmathfonts}

\usepackage{sansmathfonts}

\usepackage[mathlf,textlf]{MyriadPro}

%\usepackage[mathlf,textlf]{MyriadPro}

%\usepackage{mathpazo}

%\usepackage{mathpazo}

\usepackage[small]{eulervm}

\usepackage[small]{eulervm}

%\usepackage{newtxmath}

%\usepackage{newtxmath}

\usepackage{tabularx}

\usepackage{tabularx}

\usepackage{pbox}

\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}

\captionsetup{compatibility=false}

%\renewcommand{\arraystretch}{1}

%\renewcommand{\arraystretch}{1}

%\hypersetup{pdfpagemode={FullScreen}}

%\hypersetup{pdfpagemode={FullScreen}}

\frame{\titlepage}

\frame{\titlepage}

\section{Preliminaries}

\section{Propositional Satisfiability}

\begin{frame}

\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{frame}

\end{document}

\end{document}