Commit 7cb94e3e authored by Ralf Jung's avatar Ralf Jung

docs: complete model description

parent 82aee390
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......@@ -23,6 +23,8 @@ This definition varies slightly from the original one in~\cite{catlogic}.
An element $x \in \cofe$ of a COFE is called \emph{discrete} if
\[ \All y \in \cofe. x \nequiv{0} y \Ra x = y\]
A COFE $A$ is called \emph{discrete} if all its elements are discrete.
For a set $X$, we write $\Delta X$ for the discrete COFE with $x \nequiv{n} x' \eqdef x = x'$
\end{defn}
\begin{defn}
......@@ -31,6 +33,7 @@ This definition varies slightly from the original one in~\cite{catlogic}.
It is \emph{contractive} if
\[ \All n, x \in \cofe, y \in \cofe. (\All m < n. x \nequiv{m} y) \Ra f(x) \nequiv{n} f(x) \]
\end{defn}
The reason that contractive functions are interesting is that for every contractive $f : \cofe \to \cofe$ with $\cofe$ inhabited, there exists a fixed-point $\fix(f)$ such that $\fix(f) = f(\fix(f))$.
\begin{defn}
The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows.
......
......@@ -33,7 +33,7 @@ We start by defining the COFE of \emph{step-indexed propositions}: For every ste
\end{align*}
Now we can rewrite $\UPred(\monoid)$ as monotone step-indexed predicates over $\monoid$, where the definition of a ``monotone'' function here is a little funny.
\begin{align*}
\UPred(\monoid) \approx{}& \monoid \monra \SProp \\
\UPred(\monoid) \cong{}& \monoid \monra \SProp \\
\eqdef{}& \setComp{\pred: \monoid \nfn \SProp}{\All n, m, x, y. n \in \pred(x) \land x \mincl y \land m \leq n \land y \in \mval_m \Ra m \in \pred(y)}
\end{align*}
The reason we chose the first definition is that it is easier to work with in Coq.
......@@ -77,35 +77,35 @@ $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.
\subsection{Agreement}
Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:
\newcommand{\agc}{\mathrm{c}} % the "c" field of an agreement element
\newcommand{\agV}{\mathrm{V}} % the "V" field of an agreement element
\newcommand{\aginjc}{\mathrm{c}} % the "c" field of an agreement element
\newcommand{\aginjV}{\mathrm{V}} % the "V" field of an agreement element
\begin{align*}
\agm(\cofe) \eqdef{}& \record{\agc : \mathbb{N} \to \cofe , \agV : \SProp} \\
\agm(\cofe) \eqdef{}& \record{\aginjc : \mathbb{N} \to \cofe , \aginjV : \SProp} \\
& \text{quotiented by} \\
\melt \equiv \meltB \eqdef{}& \melt.\agV = \meltB.\agV \land \All n. n \in \melt.\agV \Ra \melt.\agc(n) \nequiv{n} \meltB.\agc(n) \\
\melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\agV \Lra m \in \meltB.\agV) \land (\All m \leq n. m \in \melt.\agV \Ra \melt.\agc(m) \nequiv{m} \meltB.\agc(m)) \\
\mval_n \eqdef{}& \setComp{\melt \in \monoid}{ n \in \melt.\agV \land \All m \leq n. \melt.\agc(n) \nequiv{m} \melt.\agc(m) } \\
\melt \equiv \meltB \eqdef{}& \melt.\aginjV = \meltB.\aginjV \land \All n. n \in \melt.\aginjV \Ra \melt.\aginjc(n) \nequiv{n} \meltB.\aginjc(n) \\
\melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\aginjV \Lra m \in \meltB.\aginjV) \land (\All m \leq n. m \in \melt.\aginjV \Ra \melt.\aginjc(m) \nequiv{m} \meltB.\aginjc(m)) \\
\mval_n \eqdef{}& \setComp{\melt \in \monoid}{ n \in \melt.\aginjV \land \All m \leq n. \melt.\aginjc(n) \nequiv{m} \melt.\aginjc(m) } \\
\mcore\melt \eqdef{}& \melt \\
\melt \mtimes \meltB \eqdef{}& (\melt.\agc, \setComp{n}{n \in \melt.\agV \land n \in \meltB.\agV \land \melt \nequiv{n} \meltB })
\melt \mtimes \meltB \eqdef{}& (\melt.\aginjc, \setComp{n}{n \in \melt.\aginjV \land n \in \meltB.\aginjV \land \melt \nequiv{n} \meltB })
\end{align*}
$\agm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$.
You can think of the $\agc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \agV$ steps.
You can think of the $\aginjc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \aginjV$ steps.
The reason we store a chain, 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 of $\agm(\cofe)$.
However, given such a chain, we cannot constructively define its limit: Clearly, the $\agV$ of the limit is the limit of the $\agV$ of the chain.
However, given such a chain, we cannot constructively define its limit: Clearly, the $\aginjV$ of the limit is the limit of the $\aginjV$ of the chain.
But what to pick for the actual data, for the element of $\cofe$?
Only if $\agV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\agV$ is smaller, the chain ``cancels'', \ie stops converging as we reach indices $n \notin \agV$.
Only if $\aginjV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\aginjV$ is smaller, the chain ``cancels'', \ie stops converging as we reach indices $n \notin \aginjV$.
To mitigate this, we apply the usual construction to close a set; we go from elements of $\cofe$ to chains of $\cofe$.
We define an injection $\ag$ into $\agm(\cofe)$ as follows:
\[ \ag(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}} \]
We define an injection $\aginj$ into $\agm(\cofe)$ as follows:
\[ \aginj(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}} \]
There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following:
\begin{mathpar}
\axiomH{ag-val}{\ag(x) \in \mval_n}
\axiomH{ag-val}{\aginj(x) \in \mval_n}
\axiomH{ag-dup}{\ag(x) = \ag(x)\mtimes\ag(x)}
\axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)}
\axiomH{ag-agree}{\ag(x) \mtimes \ag(y) \in \mval_n \Ra x \nequiv{n} y}
\axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Ra x \nequiv{n} y}
\end{mathpar}
\subsection{One-shot}
......@@ -115,17 +115,17 @@ Given some CMRA $\monoid$, we define $\oneshotm(\monoid)$ as follows:
\begin{align*}
\oneshotm(\monoid) \eqdef{}& \ospending + \osshot(\monoid) + \munit + \bot \\
\mval_n \eqdef{}& \set{\ospending, \munit} \cup \setComp{\osshot(\melt)}{\melt \in \mval_n}
\end{align*}
\begin{align*}
\mcore{\ospending} \eqdef{}& \munit & \mcore{\osshot(\melt)} \eqdef{}& \mcore\melt \\
\mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot
\end{align*}
\begin{align*}
\\%\end{align*}
%\begin{align*}
\osshot(\melt) \mtimes \osshot(\meltB) \eqdef{}& \osshot(\melt \mtimes \meltB) \\
\munit \mtimes \ospending \eqdef{}& \ospending \mtimes \munit \eqdef \ospending \\
\munit \mtimes \osshot(\melt) \eqdef{}& \osshot(\melt) \mtimes \munit \eqdef \osshot(\melt)
\end{align*}
\end{align*}%
The remaining cases of composition go to $\bot$.
\begin{align*}
\mcore{\ospending} \eqdef{}& \munit & \mcore{\osshot(\melt)} \eqdef{}& \mcore\melt \\
\mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot
\end{align*}
The step-indexed equivalence is inductively defined as follows:
\begin{mathpar}
\axiom{\ospending \nequiv{n} \ospending}
......@@ -149,34 +149,38 @@ We obtain the following frame-preserving updates:
{\osshot(\melt) \mupd \setComp{\osshot(\meltB)}{\meltB \in \meltsB}}
\end{mathpar}
%TODO: These need syncing with Coq
% \subsection{Exclusive monoid}
\subsection{Exclusive CMRA}
% Given a set $X$, we define a monoid such that at most one $x \in X$ can be owned.
% Let $\exm{X}$ be the monoid with carrier $X \uplus \{ \munit \}$ and multiplication
% \[
% \melt \cdot \meltB \;\eqdef\;
% \begin{cases}
% \melt & \mbox{if } \meltB = \munit \\
% \meltB & \mbox{if } \melt = \munit
% \end{cases}
% \]
Given a cofe $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned:
\begin{align*}
\exm(\cofe) \eqdef{}& \exinj(\cofe) + \munit + \bot \\
\mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \bot} \\
\munit \mtimes \exinj(x) \eqdef{}& \exinj(x) \mtimes \munit \eqdef \exinj(x)
\end{align*}
The remaining cases of composition go to $\bot$.
\begin{align*}
\mcore{\exinj(x)} \eqdef{}& \munit & \mcore{\munit} \eqdef{}& \munit &
\mcore{\bot} \eqdef{}& \bot
\end{align*}
The step-indexed equivalence is inductively defined as follows:
\begin{mathpar}
\infer{x \nequiv{n} y}{\exinj(x) \nequiv{n} \exinj(y)}
% The frame-preserving update
% \begin{mathpar}
% \inferH{ExUpd}
% {x \in X}
% {x \mupd \melt}
% \end{mathpar}
% is easily shown, as the only possible frame for $x$ is $\munit$.
\axiom{\munit \nequiv{n} \munit}
% Exclusive monoids are cancellative.
% \begin{proof}[Proof of cancellativity]
% If $\melt_f = \munit$, then the statement is trivial.
% If $\melt_f \neq \munit$, then we must have $\melt = \meltB = \munit$, as otherwise one of the two products would be $\mzero$.
% \end{proof}
\axiom{\bot \nequiv{n} \bot}
\end{mathpar}
$\exm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$.
We obtain the following frame-preserving update:
\begin{mathpar}
\inferH{ex-update}{}
{\exinj(x) \mupd \exinj(y)}
\end{mathpar}
%TODO: These need syncing with Coq
% \subsection{Finite Powerset Monoid}
% Given an infinite set $X$, we define a monoid $\textmon{PowFin}$ with carrier $\mathcal{P}^{\textrm{fin}}(X)$ as follows:
......
......@@ -86,13 +86,15 @@
\newcommand{\rs}{r}
\newcommand{\rsB}{s}
\newcommand{\rss}{R}
\newcommand{\pres}{\pi}
\newcommand{\wld}{w}
\newcommand{\ghostRes}{g}
%% Various pieces of syntax
\newcommand{\wsat}[4]{#1 \models_{#2} #3; #4}
\newcommand{\wsat}[3]{#1 \models_{#2} #3}
\newcommand{\wsatpre}{\textdom{pre-wsat}}
\newcommand{\wtt}[2]{#1 : #2} % well-typed term
......@@ -114,6 +116,7 @@
\newcommand{\UPred}{\textdom{UPred}}
\newcommand{\mProp}{\textdom{Prop}} % meta-level prop
\newcommand{\iProp}{\textdom{iProp}}
\newcommand{\iPreProp}{\textdom{iPreProp}}
\newcommand{\Wld}{\textdom{Wld}}
\newcommand{\Res}{\textdom{Res}}
......@@ -121,6 +124,7 @@
\newcommand{\cofeB}{U}
\newcommand{\COFEs}{\mathcal{U}} % category of COFEs
\newcommand{\iFunc}{\Sigma}
\newcommand{\fix}{\textdom{fix}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CMRA (RESOURCE ALGEBRA) SYMBOLS & NOTATION & IDENTIFIERS
......@@ -136,6 +140,8 @@
\newcommand{\melts}{A}
\newcommand{\meltsB}{B}
\newcommand{\f}{\mathrm{f}} % for "frame"
\newcommand{\mcar}[1]{|#1|}
\newcommand{\mcarp}[1]{\mcar{#1}^{+}}
\newcommand{\munit}{\varepsilon}
......@@ -321,13 +327,14 @@
% Agreement
\newcommand{\agm}{\ensuremath{\textmon{Ag}}}
\newcommand{\ag}{\textlog{ag}}
\newcommand{\aginj}{\textlog{ag}}
% Fraction
\newcommand{\fracm}{\ensuremath{\textmon{Frac}}}
% Exclusive
\newcommand{\exm}{\ensuremath{\textmon{Ex}}}
\newcommand{\exinj}{\textlog{ex}}
% Auth
\newcommand{\authm}{\textmon{Auth}}
......
......@@ -124,7 +124,7 @@ Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\t
\prop * \prop \mid
\prop \wand \prop \mid
\\&
\MU \var:\type. \pred \mid
\MU \var:\type. \term \mid
\Exists \var:\type. \prop \mid
\All \var:\type. \prop \mid
\\&
......@@ -136,7 +136,7 @@ Iris syntax is built up from a signature $\Sig$ and a countably infinite set $\t
\pvs[\term][\term] \prop\mid
\wpre{\term}[\term]{\Ret\var.\term}
\end{align*}
Recursive predicates must be \emph{guarded}: in $\MU \var. \pred$, the variable $\var$ can only appear under the later $\later$ modality.
Recursive predicates must be \emph{guarded}: in $\MU \var. \term$, the variable $\var$ can only appear under the later $\later$ modality.
Note that $\always$ and $\later$ bind more tightly than $*$, $\wand$, $\land$, $\lor$, and $\Ra$.
We will write $\pvs[\term] \prop$ for $\pvs[\term][\term] \prop$.
......
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