### logic: slight changes

parent af3ca9c5
 ... ... @@ -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$. ... ...  ... ... @@ -186,7 +186,7 @@ By conjoining the pure part of$\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 ... ...
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