diff --git a/docs/algebra.tex b/docs/algebra.tex index e9d12d74c8375b3796592523a3d54326a33cf58e..b5050387cb9299a45f6fa1066d4e3a39e90402c3 100644 --- a/docs/algebra.tex +++ b/docs/algebra.tex @@ -1,65 +1,82 @@ \section{Algebraic Structures} -\subsection{COFE} +\subsection{OFE} -The model of Iris lives in the category of \emph{Complete Ordered Families of Equivalences} (COFEs). +The model of Iris lives in the category of \emph{Ordered Families of Equivalences} (OFEs). This definition varies slightly from the original one in~\cite{catlogic}. -\begin{defn}[Chain] - Given some set $\cofe$ and an indexed family $({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \nat}$ of equivalence relations, a \emph{chain} is a function $c : \nat \to \cofe$ such that $\All n, m. n \leq m \Ra c (m) \nequiv{n} c (n)$. -\end{defn} - \begin{defn} - A \emph{complete ordered family of equivalences} (COFE) is a tuple $(\cofe, ({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \nat}, \lim : \chain(\cofe) \to \cofe)$ satisfying + An \emph{ordered family of equivalences} (OFE) is a tuple $(\ofe, ({\nequiv{n}} \subseteq \ofe \times \ofe)_{n \in \nat})$ satisfying \begin{align*} - \All n. (\nequiv{n}) ~& \text{is an equivalence relation} \tagH{cofe-equiv} \\ - \All n, m.& n \geq m \Ra (\nequiv{n}) \subseteq (\nequiv{m}) \tagH{cofe-mono} \\ - \All x, y.& x = y \Lra (\All n. x \nequiv{n} y) \tagH{cofe-limit} \\ - \All n, c.& \lim(c) \nequiv{n} c(n) \tagH{cofe-compl} + \All n. (\nequiv{n}) ~& \text{is an equivalence relation} \tagH{ofe-equiv} \\ + \All n, m.& n \geq m \Ra (\nequiv{n}) \subseteq (\nequiv{m}) \tagH{ofe-mono} \\ + \All x, y.& x = y \Lra (\All n. x \nequiv{n} y) \tagH{ofe-limit} \end{align*} \end{defn} -The key intuition behind COFEs is that elements $x$ and $y$ are $n$-equivalent, notation $x \nequiv{n} y$, if they are \emph{equivalent for $n$ steps of computation}, \ie if they cannot be distinguished by a program running for no more than $n$ steps. -In other words, as $n$ increases, $\nequiv{n}$ becomes more and more refined (\ruleref{cofe-mono})---and in the limit, it agrees with plain equality (\ruleref{cofe-limit}). -In order to solve the recursive domain equation in \Sref{sec:model} it is also essential that COFEs are \emph{complete}, \ie that any chain has a limit (\ruleref{cofe-compl}). +The key intuition behind OFEs is that elements $x$ and $y$ are $n$-equivalent, notation $x \nequiv{n} y$, if they are \emph{equivalent for $n$ steps of computation}, \ie if they cannot be distinguished by a program running for no more than $n$ steps. +In other words, as $n$ increases, $\nequiv{n}$ becomes more and more refined (\ruleref{ofe-mono})---and in the limit, it agrees with plain equality (\ruleref{ofe-limit}). \begin{defn} - 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 $x \nequiv{n} x' \eqdef x = x'$ - + An element $x \in \ofe$ of an OFE is called \emph{discrete} if + $\All y \in \ofe. x \nequiv{0} y \Ra x = y$ + An OFE $A$ is called \emph{discrete} if all its elements are discrete. + For a set $X$, we write $\Delta X$ for the discrete OFE with $x \nequiv{n} x' \eqdef x = x'$ \end{defn} \begin{defn} - A function $f : \cofe \to \cofeB$ between two COFEs is \emph{non-expansive} (written $f : \cofe \nfn \cofeB$) if - $\All n, x \in \cofe, y \in \cofe. x \nequiv{n} y \Ra f(x) \nequiv{n} f(y)$ + A function $f : \ofe \to \ofeB$ between two OFEs is \emph{non-expansive} (written $f : \ofe \nfn \ofeB$) if + $\All n, x \in \ofe, y \in \ofe. x \nequiv{n} y \Ra f(x) \nequiv{n} f(y)$ It is \emph{contractive} if - $\All n, x \in \cofe, y \in \cofe. (\All m < n. x \nequiv{m} y) \Ra f(x) \nequiv{n} f(y)$ + $\All n, x \in \ofe, y \in \ofe. (\All m < n. x \nequiv{m} y) \Ra f(x) \nequiv{n} f(y)$ \end{defn} Intuitively, applying a non-expansive function to some data will not suddenly introduce differences between seemingly equal data. Elements that cannot be distinguished by programs within $n$ steps remain indistinguishable after applying $f$. -The reason that contractive functions are interesting is that for every contractive $f : \cofe \to \cofe$ with $\cofe$ inhabited, there exists a \emph{unique} 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. + The category $\OFEs$ consists of OFEs as objects, and non-expansive functions as arrows. \end{defn} -Note that $\COFEs$ is cartesian closed. In particular: +Note that $\OFEs$ is cartesian closed. In particular: \begin{defn} - Given two COFEs $\cofe$ and $\cofeB$, the set of non-expansive functions $\set{f : \cofe \nfn \cofeB}$ is itself a COFE with + Given two OFEs $\ofe$ and $\ofeB$, the set of non-expansive functions $\set{f : \ofe \nfn \ofeB}$ is itself an OFE with \begin{align*} - f \nequiv{n} g \eqdef{}& \All x \in \cofe. f(x) \nequiv{n} g(x) + f \nequiv{n} g \eqdef{}& \All x \in \ofe. f(x) \nequiv{n} g(x) \end{align*} \end{defn} \begin{defn} - A (bi)functor $F : \COFEs \to \COFEs$ is called \emph{locally non-expansive} if its action $F_1$ on arrows is itself a non-expansive map. + A (bi)functor $F : \OFEs \to \OFEs$ is called \emph{locally non-expansive} if its action $F_1$ on arrows is itself a non-expansive map. Similarly, $F$ is called \emph{locally contractive} if $F_1$ is a contractive map. \end{defn} The function space $(-) \nfn (-)$ is a locally non-expansive bifunctor. Note that the composition of non-expansive (bi)functors is non-expansive, and the composition of a non-expansive and a contractive (bi)functor is contractive. -The reason contractive (bi)functors are interesting is that by America and Rutten's theorem~\cite{America-Rutten:JCSS89,birkedal:metric-space}, they have a unique\footnote{Uniqueness is not proven in Coq.} fixed-point. + +\subsection{COFE} + +COFEs are \emph{complete OFEs}, which means that we can take limits of arbitrary chains. + +\begin{defn}[Chain] + Given some set $\cofe$ and an indexed family $({\nequiv{n}} \subseteq \cofe \times \cofe)_{n \in \nat}$ of equivalence relations, a \emph{chain} is a function $c : \nat \to \cofe$ such that $\All n, m. n \leq m \Ra c (m) \nequiv{n} c (n)$. +\end{defn} + +\begin{defn} + A \emph{complete ordered family of equivalences} (COFE) is a tuple $(\cofe : \OFEs, \lim : \chain(\cofe) \to \cofe)$ satisfying + \begin{align*} + \All n, c.& \lim(c) \nequiv{n} c(n) \tagH{cofe-compl} + \end{align*} +\end{defn} + +\begin{defn} + The category $\COFEs$ consists of COFEs as objects, and non-expansive functions as arrows. +\end{defn} + +The function space $\ofe \nfn \cofeB$ is a COFE if $\cofeB$ is a COFE (\ie the domain $\ofe$ can actually be just an OFE). + +Completeness is necessary to take fixed-points. +For once, every \emph{contractive function} $f : \ofe \to \cofeB$ where $\cofeB$ is a COFE and inhabited has a \emph{unique} fixed-point $\fix(f)$ such that $\fix(f) = f(\fix(f))$. +Furthermore, by America and Rutten's theorem~\cite{America-Rutten:JCSS89,birkedal:metric-space}, every contractive (bi)functor from $\COFEs$ to $\COFEs$ has a unique\footnote{Uniqueness is not proven in Coq.} fixed-point. + \subsection{RA} @@ -115,7 +132,7 @@ Since Iris ensures that the global ghost state is valid, this means that we can \subsection{CMRA} \begin{defn} - A \emph{CMRA} is a tuple $(\monoid : \COFEs, (\mval_n \subseteq \monoid)_{n \in \nat},\\ \mcore{{-}}: \monoid \nfn \maybe\monoid, (\mtimes) : \monoid \times \monoid \nfn \monoid)$ satisfying: + A \emph{CMRA} is a tuple $(\monoid : \OFEs, (\mval_n \subseteq \monoid)_{n \in \nat},\\ \mcore{{-}}: \monoid \nfn \maybe\monoid, (\mtimes) : \monoid \times \monoid \nfn \monoid)$ satisfying: \begin{align*} \All n, \melt, \meltB.& \melt \nequiv{n} \meltB \land \melt\in\mval_n \Ra \meltB\in\mval_n \tagH{cmra-valid-ne} \\ \All n, m.& n \geq m \Ra \mval_n \subseteq \mval_m \tagH{cmra-valid-mono} \\ @@ -133,7 +150,7 @@ Since Iris ensures that the global ghost state is valid, this means that we can \end{align*} \end{defn} -This is a natural generalization of RAs over COFEs. +This is a natural generalization of RAs over OFEs. All operations have to be non-expansive, and the validity predicate $\mval$ can now also depend on the step-index. We define the plain $\mval$ as the limit'' of the $\mval_n$: $\mval \eqdef \bigcap_{n \in \nat} \mval_n$ @@ -209,7 +226,7 @@ Furthermore, discrete CMRAs can be turned into RAs by ignoring their COFE struct \begin{defn} The category $\CMRAs$ consists of CMRAs as objects, and monotone functions as arrows. \end{defn} -Note that every object/arrow in $\CMRAs$ is also an object/arrow of $\COFEs$. +Note that every object/arrow in $\CMRAs$ is also an object/arrow of $\OFEs$. The notion of a locally non-expansive (or contractive) bifunctor naturally generalizes to bifunctors between these categories. %TODO: Discuss how we probably have a commuting square of functors between Set, RA, CMRA, COFE. diff --git a/docs/constructions.tex b/docs/constructions.tex index 28205478fc37ac687206bcda2592aa6cc0922e02..eb72d0c87c0467463e528f9d9580d2ac79280a50 100644 --- a/docs/constructions.tex +++ b/docs/constructions.tex @@ -1,20 +1,20 @@ -\section{COFE constructions} +\section{OFE and COFE constructions} \subsection{Trivial pointwise lifting} -The COFE structure on many types can be easily obtained by pointwise lifting of the structure of the components. +The (C)OFE structure on many types can be easily obtained by pointwise lifting of the structure of the components. This is what we do for option $\maybe\cofe$, product $(M_i)_{i \in I}$ (with $I$ some finite index set), sum $\cofe + \cofe'$ and finite partial functions $K \fpfn \monoid$ (with $K$ infinite countable). \subsection{Next (type-level later)} -Given a COFE $\cofe$, we define $\latert\cofe$ as follows (using a datatype-like notation to define the type): +Given a OFE $\cofe$, we define $\latert\cofe$ as follows (using a datatype-like notation to define the type): \begin{align*} \latert\cofe \eqdef{}& \latertinj(x:\cofe) \\ \latertinj(x) \nequiv{n} \latertinj(y) \eqdef{}& n = 0 \lor x \nequiv{n-1} y \end{align*} Note that in the definition of the carrier $\latert\cofe$, $\latertinj$ is a constructor (like the constructors in Coq), \ie this is short for $\setComp{\latertinj(x)}{x \in \cofe}$. -$\latert(-)$ is a locally \emph{contractive} functor from $\COFEs$ to $\COFEs$. +$\latert(-)$ is a locally \emph{contractive} functor from $\OFEs$ to $\OFEs$. \subsection{Uniform Predicates} @@ -117,7 +117,7 @@ Notice that this core is total, as the result always lies in $\maybe\monoid$ (ra \subsection{Finite partial function} \label{sec:fpfnm} -Given some infinite countable $K$ and some CMRA $\monoid$, the set of finite partial functions $K \fpfn \monoid$ is equipped with a COFE and CMRA structure by lifting everything pointwise. +Given some infinite countable $K$ and some CMRA $\monoid$, the set of finite partial functions $K \fpfn \monoid$ is equipped with a CMRA structure by lifting everything pointwise. We obtain the following frame-preserving updates: \begin{mathpar} @@ -139,7 +139,7 @@ $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$. \subsection{Agreement} -Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows: +Given some OFE $\cofe$, we define the CMRA $\agm(\cofe)$ as follows: \begin{align*} \agm(\cofe) \eqdef{}& \set{(c, V) \in (\nat \to \cofe) \times \SProp}/\ {\sim} \-0.2em] \textnormal{where }& \melt \sim \meltB \eqdef{} \melt.V = \meltB.V \land @@ -152,7 +152,7 @@ Given some COFE \cofe, we define \agm(\cofe) as follows: \end{align*} %Note that the carrier \agm(\cofe) is a \emph{record} consisting of the two fields c and V. -\agm(-) is a locally non-expansive functor from \COFEs to \CMRAs. +\agm(-) is a locally non-expansive functor from \OFEs to \CMRAs. You can think of the c as a \emph{chain} of elements of \cofe that has to converge only for n \in V 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). @@ -175,7 +175,7 @@ There are no interesting frame-preserving updates for \agm(\cofe), but we can \subsection{Exclusive CMRA} -Given a COFE \cofe, we define a CMRA \exm(\cofe) such that at most one x \in \cofe can be owned: +Given an OFE \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) \mid \mundef \\ \mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \mundef} @@ -193,7 +193,7 @@ The step-indexed equivalence is inductively defined as follows: \axiom{\mundef \nequiv{n} \mundef} \end{mathpar} -\exm(-) is a locally non-expansive functor from \COFEs to \CMRAs. +\exm(-) is a locally non-expansive functor from \OFEs to \CMRAs. We obtain the following frame-preserving update: \begin{mathpar} diff --git a/docs/ghost-state.tex b/docs/ghost-state.tex index 95fa8e378dee01645b6518ea1cf922359dda7dec..19084064abdf675943c6751e3c6e11582cd07c2b 100644 --- a/docs/ghost-state.tex +++ b/docs/ghost-state.tex @@ -51,7 +51,7 @@ Persistence is preserved by conjunction, disjunction, separating conjunction as One of the troubles of working in a step-indexed logic is the later'' modality \later. It turns out that we can somewhat mitigate this trouble by working below the following \emph{except-0} modality: -\[ \diamond \prop \eqdef \later\FALSE \lor \Prop +$\diamond \prop \eqdef \later\FALSE \lor \prop$ This modality is useful because there is a class of assertions which we call \emph{timeless} assertions, for which we have $\timeless{\prop} \eqdef \later\prop \proves \diamond\prop$ @@ -122,11 +122,11 @@ The following rules identify the class of timeless assertions: \axiom{\timeless{\FALSE}} \infer -{\text{$\term$ or $\term'$ is a discrete COFE element}} +{\text{$\term$ or $\term'$ is a discrete OFE element}} {\timeless{\term =_\type \term'}} \infer -{\text{$\melt$ is a discrete COFE element}} +{\text{$\melt$ is a discrete OFE element}} {\timeless{\ownM\melt}} \infer diff --git a/docs/iris.sty b/docs/iris.sty index 9c5becfcd1dc6689cea6fbef6ad2e94f82506d70..51d4ebae4e1a740268817d74a126f72cd2ed5211 100644 --- a/docs/iris.sty +++ b/docs/iris.sty @@ -146,8 +146,11 @@ % List \newcommand{\List}{\ensuremath{\textdom{List}}} -\newcommand{\cofe}{T} -\newcommand{\cofeB}{U} +\newcommand{\ofe}{T} +\newcommand{\ofeB}{U} +\newcommand{\cofe}{\ofe} +\newcommand{\cofeB}{\ofeB} +\newcommand{\OFEs}{\mathcal{OFE}} % category of OFEs \newcommand{\COFEs}{\mathcal{COFE}} % category of COFEs \newcommand{\iFunc}{\Sigma} \newcommand{\fix}{\textdom{fix}} diff --git a/docs/model.tex b/docs/model.tex index fcb9d7272614f9f1981ca557a0f20394ca01f16a..f91d3d37e333c6ee9eb3cf303cc458c6cae20ae3 100644 --- a/docs/model.tex +++ b/docs/model.tex @@ -18,7 +18,7 @@ The semantic domains are interpreted as follows: \Sem{\type \to \type'} &\eqdef& \Sem{\type} \nfn \Sem{\type} \\ \end{array} \] -For the remaining base types $\type$ defined by the signature $\Sig$, we pick an object $X_\type$ in $\COFEs$ and define +For the remaining base types $\type$ defined by the signature $\Sig$, we pick an object $X_\type$ in $\OFEs$ and define $\Sem{\type} \eqdef X_\type$ diff --git a/docs/program-logic.tex b/docs/program-logic.tex index 0f3be2919805db98545ae6ff71eb6a6e07deb1fc..1d33f1f1a5e3877214fb224e7456f766c736cb63 100644 --- a/docs/program-logic.tex +++ b/docs/program-logic.tex @@ -22,16 +22,16 @@ The purpose of this section is to describe how we solve these issues. \paragraph{Picking the resources.} The key ingredient that we will employ on top of the base logic is to give some more fixed structure to the resources. -To instantiate the program logic, the user picks a family of locally contractive bifunctors $(\iFunc_i : \COFEs \to \CMRAs)_{i \in \mathcal{I}}$. +To instantiate the program logic, the user picks a family of locally contractive bifunctors $(\iFunc_i : \OFEs \to \CMRAs)_{i \in \mathcal{I}}$. (This is in contrast to the base logic, where the user picks a single, fixed CMRA that has a unit.) From this, we construct the bifunctor defining the overall resources as follows: \begin{align*} \mathcal G \eqdef{}& \nat \\ - \textdom{ResF}(\cofe^\op, \cofe) \eqdef{}& \prod_{i \in \mathcal I} \mathcal G \fpfn \iFunc_i(\cofe^\op, \cofe) + \textdom{ResF}(\ofe^\op, \ofe) \eqdef{}& \prod_{i \in \mathcal I} \mathcal G \fpfn \iFunc_i(\ofe^\op, \ofe) \end{align*} We will motivate both the use of a product and the finite partial function below. -$\textdom{ResF}(\cofe^\op, \cofe)$ is a CMRA by lifting the individual CMRAs pointwise, and it has a unit (using the empty finite partial functions). +$\textdom{ResF}(\ofe^\op, \ofe)$ is a CMRA by lifting the individual CMRAs pointwise, and it has a unit (using the empty finite partial functions). Furthermore, since the $\iFunc_i$ are locally contractive, so is $\textdom{ResF}$. Now we can write down the recursive domain equation: @@ -93,7 +93,7 @@ We can show the following properties for this form of ownership: {\ownGhost\gname{\melt : M_i} \Ra \mval_{M_i}(\melt)} \inferH{res-timeless} - {\text{$\melt$ is a discrete COFE element}} + {\text{$\melt$ is a discrete OFE element}} {\timeless{\ownGhost\gname{\melt : M_i}}} \end{mathparpagebreakable} @@ -121,8 +121,7 @@ We assume to have the following four CMRAs available: The last two are the tokens used for managing invariants, $\textmon{Inv}$ is the monoid used to manage the invariants themselves. Finally, $\textmon{State}$ is used to provide the program with a view of the physical state of the machine. -Furthermore, we assume that instances named $\gname_{\textmon{State}}$, $\gname_{\textmon{Inv}}$, $\gname_{\textmon{En}}$ and $\gname_{\textmon{Dis}}$ of these CMRAs have been created. -(We will discuss later how this assumption is discharged.) +We assume that at the beginning of the verification, instances named $\gname_{\textmon{State}}$, $\gname_{\textmon{Inv}}$, $\gname_{\textmon{En}}$ and $\gname_{\textmon{Dis}}$ of these CMRAs have been created, such that these names are globally known. \paragraph{World Satisfaction.} We can now define the assertion $W$ (\emph{world satisfaction}) which ensures that the enabled invariants are actually maintained: @@ -187,7 +186,7 @@ Fancy updates satisfy the following basic proof rules: % \inferH{fup-closeI} % {}{\knowInv\iname\prop \land \later\prop \proves \pvs[\emptyset][\set\iname] \TRUE} \end{mathparpagebreakable} -(There are no rules related to invariants here. Those rules will be discussed later, in \Sref{sec:invariants}.) +(There are no rules related to invariants here. Those rules will be discussed later, in \Sref{sec:namespaces}.) We can further define the notions of \emph{view shifts} and \emph{linear view shifts}: \begin{align*}