Commit beff0d37 authored by Martin Jonáš's avatar Martin Jonáš
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Minor changes

parent 21401a14
......@@ -119,9 +119,14 @@
\newcommand{\QFBV}{\ensuremath{\mathcal{QFBV}}}
\let\otp\titlepage
\renewcommand{\titlepage}{\otp\addtocounter{framenumber}{-1}}
\begin{document}
\frame{\titlepage}
\begin{frame}[plain]
\titlepage
\end{frame}
\section{Propositional Satisfiability}
\begin{frame}
......@@ -188,17 +193,19 @@
\bigskip
Traditionally solved by the \alert{CDCL modulo theories} -- combination of
Traditionally solved by \alert{CDCL modulo theories} -- combination of
CDCL and a specialized theory solver.
\pause
\begin{example}
Let $\varphi = (x + y = 3) \wedge (x > 1) \wedge (y = 2 \vee y = 1)$.
\begin{enumerate}[<+->]
\item Propositional model $\{ x + y = 3,~x > 1,~y = 2 \} $.
\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 $\neg (x + y = 3) \vee \neg (x > 1) \vee \neg (y = 2)$ to $\varphi$.
\item Propositional model $\{ x + y = 3,~x > 1,~y = 1 \} $.
\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}
......@@ -260,15 +267,18 @@
Quantifier bit-vector formulas traditionally solved by
\alert{quantifier instantiation}.
\pause
\begin{example}
Let $\varphi = 3 < a ~\wedge~ \forall x\,(a \not = 2 \times x)$
Let $\varphi = a > 3 ~\wedge~ \forall x\,(a \not = 2 \cdot x)$
\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 $a > 3$ 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 $3 < a ~\wedge~ (a \not = 2 \times 2)$ is satisfiable with model $a = 5$.
\item $a = 5$ is a model of $\forall x \,(a \not = 2 \times x)$.
\item $a > 3 ~\wedge~ (a \not = 2 \cdot 2)$ 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}
......@@ -319,13 +329,18 @@
Efficient algorithm are known for operations with BDDs (conjunction,
disjunction, quantifiers).
\smallskip
\bigskip
Quantification usually reduces the size of the BDD -- useful for
quantified bit-vectors.
\bigskip
Representation by (reduced and ordered) BDDs is canonical -- formula
is unsatisfiable iff the BDD has root false.
\end{frame}
\begin{frame}
We have implemented the solver Q3B, which
\begin{itemize}
\item simplifies the formula,
......@@ -334,7 +349,10 @@
too expensive.
\end{itemize}
Results show that Q3B is \alert{more efficient} than standard SMT solvers.
\bigskip
Our results and results of SMT-COMP 2016 show that Q3B is
\alert{more efficient} than standard SMT solvers.
\end{frame}
\section{Aims of the Work}
......
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