Commit 21ad2d96 authored by Ralf Jung's avatar Ralf Jung

docs: define agreement

parent f9a1045e
Pipeline #298 passed with stage
......@@ -37,7 +37,8 @@
Note that $\COFEs$ is cartesian closed.
\begin{defn}
A functor $F : \COFEs \to COFEs$ is called \emph{locally non-expansive} if its actions $F_1$ on arrows is itself a non-expansive map.
A functor $F : \COFEs \to \COFEs$ is called \emph{locally non-expansive} if its actions $F_1$ on arrows is itself a non-expansive map.
\ralf{We need bifunctors.}
Similarly, $F$ is called \emph{locally contractive} if $F_1$ is a contractive map.
\end{defn}
......@@ -156,7 +157,7 @@ Furthermore, discrete CMRAs can be turned into RAs by ignoring their COFE struct
The category $\CMRAs$ consists of CMRAs as objects, and monotone functions as arrows.
\end{defn}
Note that $\CMRAs$ is a subcategory of $\COFEs$.
The notion of a locally non-expansive (or contractive) functor naturally generalizes to functors between these categories.
The notion of a locally non-expansive (or contractive) bifunctor naturally generalizes to bifunctors between these categories.
%%% Local Variables:
......
......@@ -2,34 +2,30 @@
\section{CMRA constructions}
% We will use the notation $\mcarp{M} \eqdef |M| \setminus \{\mzero_M\}$ for the carrier of monoid $M$ without zero. When we define a carrier, a zero element is always implicitly added (we do not explicitly give it), and all cases of multiplication that are not defined (including those involving a zero element) go to that element.
% To disambiguate which monoid an element is part of, we use the notation $a : M$ to denote an $a$ s.t.\ $a \in |M|$.
% When defining a monoid, we will show some \emph{frame-preserving updates} $\melt \mupd \meltsB$ that it supports.
% Remember that
% \[
% \melt \mupd \meltsB \eqdef \always\All \melt_f. \melt \sep \melt_f \Ra \Exists \meltB \in \meltsB. \meltB \sep \melt_f.
% \]
% The rule \ruleref{FpUpd} (and, later, \ruleref{GhostUpd}) allows us to use such updates in Hoare proofs.
% The following principles generally hold for frame-preserving updates.
% \begin{mathpar}
% \infer{
% \melt \mupd \meltsB
% }{
% \melt \mupd \meltsB \cup \meltsB'
% }
% \and
% \infer{
% \melt \mupd \meltsB
% }{
% \melt \mtimes \melt_f \mupd \{ \meltB \mtimes \melt_f \mid \meltB \in \meltsB \}
% }
% \end{mathpar}
\subsection{Agreement}
\ralf{Copy some stuff from the paper, at least in case we find that there are things which are too long for the paper.}
Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:
\begin{align*}
\monoid \eqdef{}& \setComp{(c, V) \subseteq (\mathbb{N} \to T) \times \mathbb{N}}{ \All n, m. n \geq m \Ra n \in V \Ra m \in V } \\
& \text{quotiented by} \\
(c_1, V_1) \equiv (c_1, V_2) \eqdef{}& V_1 = V_2 \land \All n. n \in V_1 \Ra c_1(n) \nequiv{n} c_2(n) \\
(c_1, V_1) \nequiv{n} (c_1, V_2) \eqdef{}& (\All m \leq n. m \in V_1 \Lra m \in V_2) \land (\All m \leq n. m \in V_1 \Ra c_1(m) \nequiv{m} c_2(m)) \\
\mval_n \eqdef{}& \setComp{(c, V) \in \monoid}{ n \in V \land \All m \leq n. c(n) \nequiv{m} c(m) } \\
\mcore\melt \eqdef{}& \melt \\
\melt \mtimes \meltB \eqdef{}& (\melt.c, \setComp{n}{n \in \melt.V \land n \in \meltB.V_2 \land \melt \nequiv{n} \meltB }) \\
\melt \mdiv \meltB \eqdef{}& \melt \\
\end{align*}
$\agm(-)$ is a locally non-expansive bifunctor from $\COFEs$ to $\CMRAs$.
The reason we store a \emph{chain} $c$ of elements of $T$, rather than a single element, is that $\agm(\cofe)$ needs to be a COFE itself, so we need to be able to give a limit for every chain.
\ralf{Figure out why exactly this is not possible without adding an explicit chain here.}
There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following:
\begin{mathpar}
\axiomH{ag-dup}{\melt = \melt \mtimes \melt}
\and\axiomH{ag-agree}{\melt \mtimes \meltB \in \mval_n \Ra \melt \nequiv{n} \meltB}
\end{mathpar}
% \subsection{Exclusive monoid}
......
......@@ -69,7 +69,7 @@ For any language $\Lang$, we define the corresponding thread-pool semantics.
To instantiate Iris, you need to define the following parameters:
\begin{itemize}
\item A language $\Lang$
\item A locally contractive functor $\iFunc : \COFEs \to \CMRAs$ defining the ghost state, such that for all COFEs $A$, the CMRA $\iFunc(A)$ has a unit
\item A locally contractive bifunctor $\iFunc : \COFEs \to \CMRAs$ defining the ghost state, such that for all COFEs $A$, the CMRA $\iFunc(A)$ has a unit
\end{itemize}
\noindent
......
Markdown is supported
0%
or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment