\section{\EPbA\abstraction for \slah\ formulas}\label{app:sat-hls}
\section{{\qfpa} abstraction for \slah\ formulas}\label{app:sat-hls}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{{\EPbA} summary of $\hls{}$ atoms}
\subsection{{\qfpa} summary of $\hls{}$ atoms}
\label{ssec:sat-hls-abs}
\noindent{\bf Lemma~\ref{lem-hls}}.
...
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@@ -96,11 +96,11 @@ which can be further simplified into
%\subsection{\EPbA\ abstraction for \slah\ formulas}
\subsection{$\abs(\varphi)$: The {\qfpa} abstraction of $\varphi$}
%\label{ssec:sat-abs}
We utilize $\abs^+(\hls{}(x, y; z))$%defined in the above section
to obtain in polynomial time an equi-satisfiable \EPbA\abstraction for a symbolic heap $\varphi$, denoted by $\abs(\varphi)$.
to obtain in polynomial time an equi-satisfiable {\qfpa} abstraction for a symbolic heap $\varphi$, denoted by $\abs(\varphi)$.
%This shows that the satisfiability problem of \slah\ is in NP. Moreover, it enables using the off-the-shelf SMT solvers e.g. Z3 to solve the satisfiability problem of \slah\ formulas.
%This abstraction extends the ones proposed in~\cite{BrotherstonGK17,DBLP:journals/corr/abs-1802-05935} for ASL to deal with the inductive predicate $\hls{}$.
...
...
@@ -231,7 +231,7 @@ From the fact that the satisfiability of {\qfpa} is in NP, we conclude that the
\smallskip
\noindent{\bf Lemma~\ref{lem-eub}}.
\emph{For an {\slah} formula $\varphi\equiv\Pi: \hls{}(t'_1, t'_2; t'_3)$, a {\qfpa} formula $\xi_{eub,\varphi}(z)$ can be constructed in linear time such that for every store $s$ satisfying $s \models\abs(\varphi)$, we have $s[z \gets\eub_\varphi(s)]\models\xi_{eub,\varphi}(z)$.
\emph{For an {\slah} formula $\varphi\equiv\Pi: \hls{}(t'_1, t'_2; t'_3)$, a {\qfpa} formula $\xi_{eub,\varphi}(z)$ can be constructed in linear time such that for every store $s$ satisfying $s \models\abs(\varphi)$, we have $s[z \gets\eub_\varphi(s)]\models\xi_{eub,\varphi}(z)$ and $s[z \gets n]\not\models\xi_{eub,\varphi}(z)$ for all $n \neq\eub_\varphi(s)$.
}
\smallskip
...
...
@@ -243,7 +243,7 @@ From the construction of $\abs^+(\hls{}(t'_1, t'_2; t'_3))$ in Section~\ref{sec:
2 < t'_3\wedge2\le z \le t'_3\wedge(2\le t'_2- t'_1- z \vee t'_2- t'_1= z)\ \wedge\\
...
...
@@ -298,7 +300,8 @@ which is a {\qfpa} formula.
\noindent{\bf Lemma~\ref{lem-hls-hls}}
\emph{
Let $\varphi\equiv\Pi: \hls{}(t'_1, t'_2; t'_3)$, $\psi\equiv\hls{}(t_1, t_2; t_3)$, and $\preceq$ be a total preorder over $\addr(\varphi)\cup\addr(\psi)$ such that $C_\preceq\models t'_1 < t'_2\wedge t'_1= t_1\wedge t'_2= t_2$.
Then $\varphi\models_\preceq\psi$ iff $C_\preceq\wedge\abs(\varphi)\models\exists z.\ \xi_{eub, \varphi}(z)\wedge z \le t_3$.
Then $\varphi\models_\preceq\psi$ iff $C_\preceq\wedge\abs(\varphi)\models
\forall z.\ \xi_{eub, \varphi}(z)\rightarrow z \le t_3$.
}
\smallskip
...
...
@@ -317,18 +320,19 @@ Let $s$ be an interpretation such that $s \models C_\preceq \wedge \abs(\varphi)
At first, from $s \models\abs(\varphi)$ and Lemma~\ref{lem-eub}, we deduce that $s[z \rightarrow\eub_\varphi(s)]\models\xi_{eub,\varphi}(z)$.
%
Moreover, from the definition of $\eub_\varphi(s)$, we know that there is an unfolding scheme, say $sz_1, \cdots, sz_\ell$, such that $\eub_\varphi(s)=\max(sz_1, \cdots, sz_\ell)$. From the definition of unfolding schemes, we have $2\le sz_i \le s(t'_3)$ for every $i \in[\ell]$ and $s(t'_2)= s(t'_1)+\sum\limits_{i \in[\ell]} sz_i$. Therefore, there is a heap $h$ such that $s, h \models\varphi$ and for every $i \in[\ell]$, $h(s(t'_1)+\sum\limits_{j \in[i-1]} sz_j)= sz_i$. From the assumption $\varphi\models_\preceq\psi$, we deduce that $s, h \models\psi\equiv\hls{}(t_1, t_2; t_3)$. Then each chunk of $h$ has to be matched exactly to one unfolding of $\hls{}(t_1, t_2; t_3)$. Thus, $2\le sz_i \le s(t_3)$ for every $i \in[\ell]$. Therefore, $\eub_\varphi(s)=\max(sz_1, \cdots, sz_\ell)\le s(t_3)$. We deduce that $s[z \gets\eub_\varphi(s)]\models\xi_{eub,\varphi}(z)\wedge z \le t_3$. Consequently, $s\models\exists z.\ \xi_{eub,\varphi}(z)\wedge z \le t_3$.
Moreover, from the definition of $\eub_\varphi(s)$, we know that there is an unfolding scheme, say $sz_1, \cdots, sz_\ell$, such that $\eub_\varphi(s)=\max(sz_1, \cdots, sz_\ell)$. From the definition of unfolding schemes, we have $2\le sz_i \le s(t'_3)$ for every $i \in[\ell]$ and $s(t'_2)= s(t'_1)+\sum\limits_{i \in[\ell]} sz_i$. Therefore, there is a heap $h$ such that $s, h \models\varphi$ and for every $i \in[\ell]$, $h(s(t'_1)+\sum\limits_{j \in[i-1]} sz_j)= sz_i$. From the assumption $\varphi\models_\preceq\psi$, we deduce that $s, h \models\psi\equiv\hls{}(t_1, t_2; t_3)$. Then each chunk of $h$ has to be matched exactly to one unfolding of $\hls{}(t_1, t_2; t_3)$. Thus, $2\le sz_i \le s(t_3)$ for every $i \in[\ell]$. Therefore, $\eub_\varphi(s)=\max(sz_1, \cdots, sz_\ell)\le s(t_3)$. We deduce that $s[z \gets\eub_\varphi(s)]\models\xi_{eub,\varphi}(z)\wedge z \le t_3$. Moreover, from Lemma~\ref{lem-eub}, we know that for all $n \neq\eub_\varphi(s)$, $s[z \gets n]\not\models\xi_{eub,\varphi}(z)$. Consequently, $s \models\forall z.\ \xi_{eub,\varphi}(z)\rightarrow z \le t_3$.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\emph{``If'' direction}: Suppose $C_\preceq\wedge\abs(\varphi)\models\exists z.\ \xi_{eub, \varphi}(z)\wedge z \le t_3$ and $s, h \models C_\preceq\wedge\Pi: \hls{}(t'_1, t'_2; t'_3)$. We show $s, h \models\psi$.
\noindent\emph{``If'' direction}: Suppose $C_\preceq\wedge\abs(\varphi)\models\forall z.\ \xi_{eub, \varphi}(z)\rightarrow z \le t_3$ and $s, h \models C_\preceq\wedge\Pi: \hls{}(t'_1, t'_2; t'_3)$. We show $s, h \models\psi$.
From $s, h \models C_\preceq\wedge\Pi: \hls{}(t'_1, t'_2; t'_3)$, we know that there are $sz_1, \cdots, sz_\ell$ such that they are the sequence of chunk sizes in $h$. Therefore, $(sz_1, \cdots, sz_\ell)$ is an unfolding scheme of $\varphi$ w.r.t. $s$. From the definition of $\eub_\varphi(s)$, we have $\eub_\varphi(s)$ is the maximum chunk size upper bound of the unfolding schemes of $\varphi$ w.r.t. $s$. Therefore, $\max(sz_1, \cdots, sz_\ell)\le\eub_\varphi(s)$.
From $s, h \models C_\preceq\wedge\Pi: \hls{}(t'_1, t'_2; t'_3)$, we deduce that $s \models C_\preceq\wedge\abs(\varphi)$.
Because $C_\preceq\wedge\abs(\varphi)\models\exists z.\ \xi_{eub, \varphi}(z)\wedge z \le t_3$, we have $s \models\exists z.\ \xi_{eub, \varphi}(z)\wedge z \le t_3$.
Then $s[z \gets n]\models\xi_{eub, \varphi}(z)\wedge z \le t_3$ for some $n \in\NN$.
Thus $s[z \gets n]\models\xi_{eub, \varphi}(z)$. Moreover, from Lemma~\ref{lem-eub}, we know that $s[z \gets\eub_\varphi(s)]\models\xi_{eub, \varphi}(z)$.
According to the definition of $\eub_\varphi(s)$, we deduce that $n =\eub_\varphi(s)$. Therefore, $\max(sz_1, \cdots, sz_\ell)\le\eub_\varphi(s)\le s(t_3)$. It follows that for every $i \in[\ell]$, $2\le sz_i \le s(t_3)$. We conclude that $s, h \models\hls{}(t_1, t_2; t_3)\equiv\psi$.
Because $C_\preceq\wedge\abs(\varphi)\models\forall z.\ \xi_{eub, \varphi}(z)\rightarrow z \le t_3$, we have $s \models\forall z.\ \xi_{eub, \varphi}(z)\rightarrow z \le t_3$.
%Then $s[z \gets n] \models \xi_{eub, \varphi}(z) \wedge z \le t_3$ for some $n \in \NN$.
From Lemma~\ref{lem-eub}, we know that $s[z \gets\eub_\varphi(s)]\models\xi_{eub, \varphi}(z)$.
Therefore, we deduce that $s[z \gets\eub_\varphi(s)]\models z \le t_3$, namely, $\eub_\varphi(s)\le s(t_3)$. Then $\max(sz_1, \cdots, sz_\ell)\le\eub_\varphi(s)\le s(t_3)$. It follows that for every $i \in[\ell]$, $2\le sz_i \le s(t_3)$. We conclude that $s, h \models\hls{}(t_1, t_2; t_3)\equiv\psi$.
@@ -89,7 +89,7 @@ with respect to stacks %\mihaela{r3: not clear}
can be captured by a {\qfpa} formula.
\begin{lemma}\label{lem-eub}
For an {\slah} formula $\varphi\equiv\Pi: \hls{}(t'_1, t'_2; t'_3)$, a {\qfpa}formula $\xi_{eub,\varphi}(z)$ can be constructed in linear time such that for every store $s$ satisfying $s \models\abs(\varphi)$, we have $s[z \gets\eub_\varphi(s)]\models\xi_{eub,\varphi}(z)$.
For an {\slah} formula $\varphi\equiv\Pi: \hls{}(t'_1, t'_2; t'_3)$, a {\qfpa} formula $\xi_{eub,\varphi}(z)$ can be constructed in linear time such that for every store $s$ satisfying $s \models\abs(\varphi)$, we have $s[z \gets\eub_\varphi(s)]\models\xi_{eub,\varphi}(z)$and $s[z \gets n]\not\models\xi_{eub,\varphi}(z)$ for all $n \neq\eub_\varphi(s)$.
\end{lemma}
The following lemma (proof in the appendix) provides the correct test used
@@ -47,20 +47,20 @@ 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
an{\EPbA} abstraction of a \slah\ formula $\varphi$, denoted by $\abs(\varphi)$,
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 {\EPbA}, $t_1= t_2\vee(t_1 < t_2\wedge\abs^+(\hls{}(t_1, t_2; t_3)))$.
\item Lastly, the separating conjunction is encoded by an {\EPbA} formula constraining the address terms of spatial atoms.
\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.
%
The crux of this encoding and
its originality with respect to the ones proposed for ASL in~\cite{BrotherstonGK17}
is the computation of $\abs^+(\hls{}(t_1, t_2; t_3))$,
which are the least-fixed-point summaries in \EPbA\for $\hls{}(t_1, t_2; t_3)$.
which are the least-fixed-point summaries in {\qfpa} for $\hls{}(t_1, t_2; t_3)$.
In the sequel, we show how to compute them.
%Moreover, $\abs(\varphi)$ will be used as a basic ingredient of the entailment procedure in Section~\ref{sec:ent}.
