Commit beff0d37 by Martin Jonáš

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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