logic: slight changes

atva2021/logic-hls.tex

@@ -20,7 +20,7 @@ and symbolic heaps $\varphi$ is given by the following grammar:
 \begin{array}{l l l r}
 t & ::= x \mid n \mid t+t &\ \ \ & \mbox{terms}
 \\ 
-\Pi & ::= t=t \mid t \ne t \mid t \leq t \mid t < t \mid \Pi \land \Pi &\ \ \  & \mbox{pure formulas}
+\Pi & ::= \top \mid \bot \mid t=t \mid t \ne t \mid t \leq t \mid t < t \mid \Pi \land \Pi &\ \ \  & \mbox{pure formulas}
 \\ 
 \Sigma & ::= \emp \mid t \pto t \mid \blk(t, t) \mid \hls{}(t, t; t^\infty) \mid \Sigma \sepc \Sigma &\ \ \  & \mbox{spatial formulas}
 \\ 
@@ -42,7 +42,7 @@ the rules in Equations~(\ref{eq:hlsv-emp}) and (\ref{eq:hlsv-rec}),
 where $v$ is a variable interpreted over $\NN\cup\{\infty\}$. 
 %Here we assume that $c \ge 2$ since $x \pto x'-x \sepc \blk(x+1,x')$ occurs in the body of Rule~(\ref{eq:hlsv-rec}).
 An atom $\hls{}(x,y;\infty)$ is also written $\hls{}(x,y)$;
-$\top$ and $\bot$ are shorthands for $x=x$ and $x\ne x$ atoms..
+%$\top$ and $\bot$ are shorthands for $x=x$ and $x\ne x$ atoms..
 Whenever one of $\Pi$ or $\Sigma$ is empty, we omit the colon. 
 We write $\fv(\varphi)$ for the set of free variables occurring in $\varphi$. 
 If $\varphi = \exists\vec{z}\cdot\Pi:\Sigma$, 
@@ -92,6 +92,7 @@ The satisfaction relation $s,h\models \varphi$,
 	where $s$ is a stack, $h$ a heap, and $\varphi$ a \slah\ formula,
 	is defined by: % structural induction on $\varphi$ as follows:
 \begin{itemize}
+\item $s,h \models  \top$ always and never $s,h \models  \bot$,
 \item $s,h \models  t_1 \sim t_2$ iff   
 	$s(t_1) \sim s(t_2)$, where $\sim\in\{=,\ne,\le,<\}$,
 %
@@ -138,7 +139,7 @@ $s,h \models  \exists\vec{z}\cdot\Pi:\Sigma$ iff $\exists \vec{n}\in\NN^{|\vec{z
 \end{definition}
 %\vspace{-2mm}
 
-We write $A \models B$ for $A$ and $B$ sub-formula in \slah\ to mean
+We write $A \models B$ for $A$ and $B$ (sub-)formula in \slah\ to mean
 that $A$ entails $B$, i.e., that for any model $(s,h)$ such that $s,h\models A$
 then $s,h\models B$.
 

atva2021/overview.tex

@@ -186,7 +186,7 @@ By conjoining the pure part of $\texttt{path}_{\texttt{4-9}}$ with
 formulas $\texttt{pb}^\Sigma_{\texttt{4-9}}$ and 
          $\texttt{pb}^\sepc_{\texttt{4-9}}$,
 we obtain an equi-satisfiable existentially quantified {\PbA} formula
- whose satisfiability is NP-complete.
+ whose satisfiability is a NP-complete problem.
 
 \smallskip
 \noindent