Few changes to slides.

Obhajoba/slides.tex

-\documentclass[aspectratio=169]{beamer}
+%\documentclass[aspectratio=169]{beamer}
+\documentclass{beamer}
 %\usepackage[czech]{babel}
 \usepackage[utf8]{inputenc}
 %balíček na A4
@@ -128,58 +129,6 @@
  \titlepage
 \end{frame}
 
-\section{Propositional Satisfiability}
-\begin{frame}
-  \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}
@@ -189,33 +138,32 @@
 
   \smallskip
 
-  Function symbols are given interpretations by the theory.
+  Function and relation symbols are given interpretations by the theory.
 
   \bigskip
 
-  Traditionally solved by \alert{CDCL modulo theories} -- combination of
-  CDCL and a specialized theory solver.
-
-  \pause
-
-  \begin{example}
-    Let $\varphi = (x + y = 3) \wedge (x > 1) \wedge (y = 2 \vee y = 1)$.
+  For example
+  \begin{itemize}
+  \item $x + y = 3~\wedge~x > 1~\wedge~y > x$ is satisfiable over the
+    integers,
+  \item $3 = x + x$ is unsatisfiable over the integers,
+  \item $3 = x + x$ is satisfiable over the rationals.
+  \end{itemize}
 
-    \begin{enumerate}[<+->]
-    \item Propositional model $\{ x + y = 3,~x > 1,~y = 2,~\neg(y = 1) \} $.
-    \item This model \textbf{is not} consistent with the theory of integers.
-    \item Conjoin $\neg (x + y = 3) \vee \neg (x > 1) \vee \neg (y = 2) \vee (y = 1)$ to $\varphi$.
-    \item Propositional model $\{ x + y = 3,~x > 1,~\neg(y=2),~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}
 
+  Hardware and programming languages do not work over integers.
+
+  The formula
+  \[
+    (x > 0)~\wedge~(y > 0)~\wedge~(x + y < 0)
+  \]
+  should be satisfiable due to overflows.
+
   Theory of bit-vectors describes bounded integers (or vectors of
   bits) with:
   \begin{itemize}
@@ -224,16 +172,9 @@
   \item comparison.
   \end{itemize}
 
-  Often used in modelling of software.
-
-  \bigskip
-
-  Different from mathematical integers -- for example the formula
-  \[
-    \varphi = (x^{[32]} >_s 0)~\wedge~(y^{[32]} >_s 0)~\wedge~(x^{[32]} + y^{[32]} = 0)
-  \]
+  \smallskip
 
-  is \alert{satisfiable}.
+  Often used in modelling of software.
 \end{frame}
 
 \section{Quantifier-free Bit-Vector Logic}
@@ -267,21 +208,6 @@
 
   Quantifier bit-vector formulas traditionally solved by
   \alert{quantifier instantiation}.
-
-  \pause
-
-  \begin{example}
-    Let $\varphi = a > 3 ~\wedge~ \forall x\,(a \not = 2 \cdot x)$
-    \pause
-    \begin{itemize}[<+->]
-    \item $a > 3$ is satisfiable with model $a = 4$.
-    \item $a = 4$ not a model of $\forall x \,(a \not = 2 \cdot x)$ -- consider $x = 2$.
-    \item Add instance of the quantifier for $x = 2$.
-    \item $a > 3 ~\wedge~ (a \not = 2 \cdot 2)$ is satisfiable with model $a = 5$.
-    \item $a = 5$ is a model of $\forall x \,(a \not = 2 \cdot x)$.
-    \item $\varphi$ is satisfiable
-    \end{itemize}
-  \end{example}
 \end{frame}
 
 \section{Symbolic Approach to Quantified Bit-Vectors}
@@ -341,7 +267,9 @@
 \end{frame}
 
 \begin{frame}
-  We have implemented the solver Q3B, which
+  \frametitle{Symbolic Approach to Quantified Bit-Vectors}
+
+  I have implemented the solver Q3B, which
   \begin{itemize}
   \item simplifies the formula,
   \item converts the formula to the BDD,
@@ -351,8 +279,7 @@
 
   \bigskip
 
-  Our results and results of SMT-COMP 2016 show that Q3B is
-  \alert{more efficient} than standard SMT solvers.
+  Q3B is the \alert{winner} of SMT Competition 2016 in the category of quantified bit-vectors.
 \end{frame}
 
 \section{Aims of the Work}
@@ -371,7 +298,8 @@
   \begin{itemize}
   \item Determining a computational complexity of quantified bit-vector logic with
     bit-widths represented in binary.
-  \item Extending simplifications using unconstrained variables
+  \item Extending known syntactic simplifications of quantified
+    bit-vector formulas.
   \end{itemize}
 \end{frame}