Commit 209dfbb6 by Mihaela SIGHIREANU

### sketch of coNP for entail

parent d604fec2
 %!TEX root = cade-28.tex \section{Complexity of the entailment problem} Theorem~\ref{thm-entail} states that the upper bound of the entailment problem is \textsc{EXPTIME}. The lower bound is given by the complexity of the entailment problem for quantifier free formulas in \textsc{ASL}, which is shown to be \textsc{coNP} in~\cite{BrotherstonGK17}. \textbf{Claim:} The entailment problem in \slah\ is in \textbf{coNP}. \begin{proof} Let fix $\varphi \models \psi$ an entailment problem in \slah. We build in polynomial time a \PbA\ formula $\chi$ from $\varphi$ and $\psi$ such that for any stack $s$, $s\models \chi$ iff $\exists h.~s,h\models \varphi$ and $s,h \not\models \psi$. We suppose that both formula $\varphi$ and $\psi$ have been transformed in normal form as follows: \begin{itemize} \item atoms $\hls{}(x,y;v')$ with $\abs(\varphi)\models (v'<2 \lor x=y)$ have been removed and replaced by $x=y$\mihaela{use $\xi$ ?} \item atoms $\hls{}(x,y;v')$ with $\abs(\varphi)\models v'=2$ have been replaced by $y-x=2 : x\pto 2 \star \blk(x+1,y)$ \item ...\mihaela{TODO} \end{itemize} The normalization can be done in a linear number of calls to the \EPbA\ procedure. The formula $\chi$ shall ensure that $\varphi$ is satisfiable, so it contains as conjunct $\abs(\varphi)$. The entailment may be invalid iff one of the following holds: \begin{itemize} \item $\psi$ is not satisfiable, i.e., $\abs(\psi)$ is not satisfiable. \item there exists an address inside a spatial atom of $\varphi$ which is not inside a spatial atom of $\psi$; we denote this formula by $cov(\varphi,\psi)$; \item similarly, $cov(\psi,\varphi)$ (the semantics is non intuitionistic), \item a points-to or a heap-list atom of $\psi$ covers'' a domain of addresses which is inside the one of a block atom of $\varphi$; we denote this formula by $nblk(\varphi,\psi)$; \item a points-to atom of $\psi$ defines an address inside (strictly) a heap-list atom of $\varphi$; we denote this formula by $npto(\varphi,\psi)$; \item a points-to atom $x\pto v$ of $\psi$ defines the same address as the start address of a heap-list atom $\hls{}(x,y;v')$ in $\varphi$; we denote this formula by $npto'(\varphi,\psi)$; (this is true because we are in the normal form); \item a points-to atoms $x\pto v$ of $\psi$ defines the same address as a points-to atom $x\pto v'$ of $\varphi$ but $v\neq v'$; \item a heap-list atom $\hls{}(x,y;v')$ of $\psi$ starts at an address at which is defined a points-to atom $x\pto v$ of $\varphi$ such that $v> v'$; \item a heap-list atom $\hls{}(x,y;v')$ of $\psi$ covers'' the domain of addresses if an atom $\hls{}(z,w;v)$ of $\varphi$ but $v' < v$; \item a heap-list atom $\hls{}(x,y;v')$ of $\psi$ covers'' a domain of addresses that can not be filded into a $\hls{}$ atom with chunks of maximal size less or equal to $v'$\mihaela{difficult in P!} \end{itemize} \end{proof}
 ... ... @@ -88,6 +88,7 @@ Zhilin Wu\inst{2} \orcidID{0000-0003-0899-628X} \appendix %\section{Appendixes} \input{app-sat-hls.tex} \input{app-ent-hls.tex} %% temporary %\newpage ... ...
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