@@ -42,7 +42,7 @@ Now we can rewrite $\UPred(\monoid)$ as monotone step-indexed predicates over $\
The reason we chose the first definition is that it is easier to work with in Coq.
\clearpage
\section{CMRA constructions}
\section{RA and CMRA constructions}
\subsection{Product}
\label{sec:prodm}
...
...
@@ -378,54 +378,79 @@ We obtain the following frame-preserving update:
% }
% \end{mathpar}
% \subsection{STS with tokens monoid}
% \label{sec:stsmon}
\subsection{STS with tokens}
\label{sec:stsmon}
% Given a state-transition system~(STS) $(\STSS, \ra)$, a set of tokens $\STSS$, and a labeling $\STSL: \STSS \ra \mathcal{P}(\STST)$ of \emph{protocol-owned} tokens for each state, we construct a monoid modeling an authoritative current state and permitting transitions given a \emph{bound} on the current state and a set of \emph{locally-owned} tokens.
Given a state-transition system~(STS, \ie a directed graph) $(\STSS, {\stsstep}\subseteq\STSS\times\STSS)$, a set of tokens $\STST$, and a labeling $\STSL: \STSS\ra\wp(\STST)$ of \emph{protocol-owned} tokens for each state, we construct a monoid modeling an authoritative current state and permitting transitions given a \emph{bound} on the current state and a set of \emph{locally-owned} tokens.
% The construction follows the idea of STSs as described in CaReSL \cite{caresl}.
% We first lift the transition relation to $\STSS \times \mathcal{P}(\STST)$ (implementing a \emph{law of token conservation}) and define upwards closure:
% \begin{align*}
% (s, T) \ra (s', T') \eqdef&\, s \ra s' \land \STSL(s) \uplus T = \STSL(s') \uplus T' \\
% \upclose(S, T) \eqdef&\, \setComp{ s' \in \STSS}{\exists s \in S.\; \textsf{frame}(s, T) \ststrans \textsf{frame}(s', T) }
% \end{align*}
The construction follows the idea of STSs as described in CaReSL \cite{caresl}.
We first lift the transition relation to $\STSS\times\wp(\STST)$ (implementing a \emph{law of token conservation}) and define a stepping relation for the \emph{frame} of a given token set:
\begin{align*}
(s, T) \stsstep (s', T') \eqdef{}& s \stsstep s' \land\STSL(s) \uplus T = \STSL(s') \uplus T' \\
s \stsfstep{T} s' \eqdef{}&\Exists T_1, T_2. T_1 \sep\STSL(s) \cup T \l+and (s, T_1) \stsstep (s', T_2)
\end{align*}
% \noindent
% We have
% \begin{quote}
% If $(s, T) \ra (s', T')$\\
% and $T_\f \sep (T \uplus \STSL(s))$,\\
% then $\textsf{frame}(s, T_\f) \ra \textsf{frame}(s', T_\f)$.
% \end{quote}
% \begin{proof}
% This follows directly by framing the tokens in $\STST \setminus (T_\f \uplus T \uplus \STSL(s))$ around the given transition, which yields $(s, \STST \setminus (T_\f \uplus \STSL{T}(s))) \ra (s', T' \uplus (\STST \setminus (T_\f \uplus T \uplus \STSL{T}(s))))$.
% This is exactly what we have to show, since we know $\STSL(s) \uplus T = \STSL(s') \uplus T'$.
% \end{proof}
We further define \emph{closed} sets of states (given a particular set of tokens) as well as the \emph{closure} of a set:
\begin{align*}
\STSclsd(S, T) \eqdef{}&\All s \in S. \STSL(s) \sep T \land\All s'. s \stsfstep{T} s' \Ra s' \in S \\
\upclose(S, T) \eqdef{}&\setComp{ s' \in\STSS}{\Exists s \in S. s \stsftrans{T} s' }
\end{align*}
% Let $\STSMon{\STSS}$ be the monoid with carrier
% \[
% \setComp{ (s, S, T) \in \exm{\STSS} \times \mathcal{P}(\STSS) \times \mathcal{P}(\STST) }{ \begin{aligned} &(s = \munit \lor s \in S) \land \upclose(S, T) = S \land{} \\& S \neq \emptyset \land \All s \in S. \STSL(s) \sep T \end{aligned} }
% Some sugar makes it more convenient to assert being at least in a certain state and owning some tokens: $(s, T) : \STSMon{\STSS} \eqdef (\munit, \upclose(\{s\}, T), T) : \STSMon{\STSS}$, and
% Assume some upwards-closed $S_\f, T_\f$ (the frame cannot be authoritative) s.t.\ $s \in S_\f$ and $T_\f \sep (T \uplus \STSL(s))$. We have to show that this frame combines with our final monoid element, which is the case if $s' \in S_\f$ and $T_\f \sep T'$.
% By upward-closedness, it suffices to show $\textsf{frame}(s, T_\f) \ststrans \textsf{frame}(s', T_\f)$.
% This follows by induction on the path $(s, T) \ststrans (s', T')$, and using the lemma proven above for each step.
The core of the STS construction is only satisfying the RA axioms because we are \emph{not} demanding the core to be a homomorphism---all we demand is for the core to be monotone with respect the \ruleref{ra-incl}.
In other words, the following does \emph{not} hold for the STS core as defined above:
