Commit 5749735d by ehilin02

### abs from appendix to main text

parent f9e07fd5
 ... ... @@ -96,6 +96,8 @@ which can be further simplified into %Moved to the main text \hide{ \subsection{$\abs(\varphi)$: The {\qfpa} abstraction of $\varphi$} %\label{ssec:sat-abs} ... ... @@ -227,6 +229,7 @@ From the fact that the satisfiability of {\qfpa} is in NP, we conclude that the %The formulas $\exists k.\ k \ge 1 \wedge 2k = y - x$ in $\abs^+(\hls{}(x, y; z))$ can be equivalently replaced by $y -x > 0 \wedge y -x \equiv 0 \bmod 2$. Therefore, $\abs(\varphi)$ is essentially a quantifier-free \PbA\ formula containing modulo constraints. The satisfiability of such formulas is still NP-complete. %From now on, we shall assume that {\bf $\abs(\varphi)$ is a quantifier-free \PbA\ formula containing modulo constraints}. \end{remark} } \section{Proof of Lemma~\ref{lem-eub}} ... ...
 ... ... @@ -47,20 +47,77 @@ We recall that the satisfiability problem of {\qfpa} and {\EPbA} is NP-complete. %ASL formulas into equi-satisfiable {\EPbA} formulas. We basically follow the same idea as ASL to build a {\qfpa} abstraction of a \slah\ formula $\varphi$, denoted by $\abs(\varphi)$, that encodes its satisfiability: \begin{compactitem} \item At first, points-to atoms $t_1 \pto t_2$ are transformed into $\blk(t_1, t_1+1)$. \item Then, the block atoms $\blk(t_1, t_2)$ are encoded by the constraint $t_1 < t_2$. \item The predicate atoms $\hls{}(t_1, t_2; t_3)$, absent in ASL, are encoded by a formula in {\qfpa}, $t_1 = t_2 \vee (t_1 < t_2 \wedge \abs^+(\hls{}(t_1, t_2; t_3)))$. \item Lastly, the separating conjunction is encoded by an {\qfpa} formula constraining the address terms of spatial atoms. \end{compactitem} The Appendix~\ref{app:sat-hls} provides more details. an equi-satisfiable {\qfpa} abstraction of a \slah\ formula $\varphi$. %that encodes its satisfiability: %\begin{compactitem} %\item At first, points-to atoms $t_1 \pto t_2$ are transformed into $\blk(t_1, t_1+1)$. %\item Then, the block atoms $\blk(t_1, t_2)$ are encoded by the constraint $t_1 < t_2$. %\item The predicate atoms $\hls{}(t_1, t_2; t_3)$, absent in ASL, are encoded by a formula in {\qfpa}, $t_1 = t_2 \vee (t_1 < t_2 \wedge \abs^+(\hls{}(t_1, t_2; t_3)))$. %\item Lastly, the separating conjunction is encoded by an {\qfpa} formula constraining the address terms of spatial atoms. %\end{compactitem} %We utilize $\abs^+(\hls{}(x, y; z))$ %defined in the above section %to obtain in polynomial time an equi-satisfiable {\qfpa} abstraction for a symbolic heap $\varphi$, denoted by $\abs(\varphi)$. We introduce some notations first. % Given a formula $\varphi\equiv \Pi : \Sigma$, $\atoms(\varphi)$ denotes the set of spatial atoms in $\Sigma$, and $\patoms(\varphi)$ denotes the set of predicate atoms in $\Sigma$. We also denote $\overline{\patoms}(\varphi)$ for $\atoms(\varphi)\setminus\patoms(\varphi)$. \begin{definition}{(Presburger abstraction of \slah\ formula)} Let $\varphi\equiv \Pi : \Sigma$ be a \slah\ formula. The abstraction of $\varphi\equiv \Pi : \Sigma$, denoted by $\abs(\varphi)$, is the formula $\Pi\wedge\phi_{\Sigma}\wedge\phi_*$ where: \begin{itemize} \item $\phi_{\Sigma}\triangleq\bigwedge\limits_{a \in \atoms(\varphi)}\abs(a)$ such that {\small \begin{eqnarray} \abs(t_1\mapsto t_2) & \triangleq & \ltrue \\ \abs(\blk(t_1,t_2)) & \triangleq & t_1
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