section 5

atva2021/entail-hls-one-atom.tex

@@ -86,10 +86,10 @@ Let $\varphi \equiv x < y: \hls{}(x, y; 3)$ and $s$ be a store such that $s(x)=
 The following lemma (proved in the appendix) states that 
 the effective upper bounds of chunks in heap lists atoms of $\varphi$ 
 with respect to stacks %\mihaela{r3: not clear}
-can be captured by an {\EPbA} formula.
+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  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 $s[z \gets \eub_\varphi(s)] \models \xi_{eub,\varphi}(z)$. 
 \end{lemma}
 
 The following lemma (proof in the appendix) provides the correct test used 
@@ -101,10 +101,11 @@ Then $\varphi \models_\preceq \psi$ iff
 %$C_\preceq \wedge \abs(\varphi)  \models  \exists z.\ \xi_{eub, \varphi}(z) \wedge z \le t_3$.  
 $C_\preceq \wedge \abs(\varphi)  \models  \forall z.\ \xi_{eub, \varphi}(z) \rightarrow z \le t_3$.  
 \end{lemma}
-From Lemma~\ref{lem-hls-hls}, it follows that the entailment of 
+From Lemma~\ref{lem-hls-hls}, it follows that $\varphi \equiv \Pi: \hls{}(t'_1, t'_2; t'_3) \models_\preceq \hls{}(t_1, t_2; t_3)$ is invalid iff $C_\preceq \wedge \abs(\varphi)  \wedge  \exists z.\ \xi_{eub, \varphi}(z) \wedge \neg z \le t_3$ is satisfiable, which is an {\EPbA} formula. Therefore, this special case of the ordered entailment problem is in coNP.
 
+\medskip
 %\vspace{-4mm}
-\subsubsection{At least two atoms in the antecedent:} Recall that 
+\noindent {\bf At least two atoms in the antecedent:} Recall that 
 $\varphi\equiv C_\preceq \wedge \Pi: a_1 \sepc \cdots \sepc a_m$; 
 a case analysis on the form of the first atom of the antecedent, $a_1$, follows.