constructions.tex 18.6 KB
 Ralf Jung committed Mar 11, 2016 1 \section{COFE constructions}  Ralf Jung committed Jan 31, 2016 2   Ralf Jung committed Oct 18, 2016 3 4 5 6 7 \subsection{Trivial pointwise lifting} The COFE 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).  Ralf Jung committed Mar 11, 2016 8 9 \subsection{Next (type-level later)}  Ralf Jung committed Jul 27, 2016 10 Given a COFE $\cofe$, we define $\latert\cofe$ as follows (using a datatype-like notation to define the type):  Ralf Jung committed Mar 11, 2016 11 \begin{align*}  Ralf Jung committed Mar 23, 2016 12  \latert\cofe \eqdef{}& \latertinj(x:\cofe) \\  Ralf Jung committed Mar 11, 2016 13 14  \latertinj(x) \nequiv{n} \latertinj(y) \eqdef{}& n = 0 \lor x \nequiv{n-1} y \end{align*}  Ralf Jung committed Mar 23, 2016 15 16 Note that in the definition of the carrier $\latert\cofe$, $\latertinj$ is a constructor (like the constructors in Coq), \ie this is short for $\setComp{\latertinj(x)}{x \in \cofe}$.  Ralf Jung committed Mar 11, 2016 17 18 $\latert(-)$ is a locally \emph{contractive} functor from $\COFEs$ to $\COFEs$.  Ralf Jung committed Mar 23, 2016 19   Ralf Jung committed Mar 11, 2016 20 21 22 23 \subsection{Uniform Predicates} Given a CMRA $\monoid$, we define the COFE $\UPred(\monoid)$ of \emph{uniform predicates} over $\monoid$ as follows: \begin{align*}  Robbert Krebbers committed Oct 17, 2016 24  \UPred(\monoid) \eqdef{} \setComp{\pred: \nat \times \monoid \to \mProp}{  Ralf Jung committed Mar 11, 2016 25 26 27 28 29 30 31 32 33 34  \begin{inbox}[c] (\All n, x, y. \pred(n, x) \land x \nequiv{n} y \Ra \pred(n, y)) \land {}\\ (\All n, m, x, y. \pred(n, x) \land x \mincl y \land m \leq n \land y \in \mval_m \Ra \pred(m, y)) \end{inbox} } \end{align*} where $\mProp$ is the set of meta-level propositions, \eg Coq's \texttt{Prop}. $\UPred(-)$ is a locally non-expansive functor from $\CMRAs$ to $\COFEs$. One way to understand this definition is to re-write it a little.  Ralf Jung committed Mar 23, 2016 35 We start by defining the COFE of \emph{step-indexed propositions}: For every step-index, the proposition either holds or does not hold.  Ralf Jung committed Mar 11, 2016 36 \begin{align*}  Robbert Krebbers committed Oct 17, 2016 37 38  \SProp \eqdef{}& \psetdown{\nat} \\ \eqdef{}& \setComp{X \in \pset{\nat}}{ \All n, m. n \geq m \Ra n \in X \Ra m \in X } \\  Ralf Jung committed Mar 14, 2016 39  X \nequiv{n} Y \eqdef{}& \All m \leq n. m \in X \Lra m \in Y  Ralf Jung committed Mar 11, 2016 40 \end{align*}  Ralf Jung committed Mar 23, 2016 41 Notice that this notion of $\SProp$ is already hidden in the validity predicate $\mval_n$ of a CMRA:  Ralf Jung committed Mar 14, 2016 42 We could equivalently require every CMRA to define $\mval_{-}(-) : \monoid \nfn \SProp$, replacing \ruleref{cmra-valid-ne} and \ruleref{cmra-valid-mono}.  Ralf Jung committed Mar 14, 2016 43   Ralf Jung committed Mar 11, 2016 44 45 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*}  Ralf Jung committed Mar 12, 2016 46  \UPred(\monoid) \cong{}& \monoid \monra \SProp \\  Ralf Jung committed Mar 11, 2016 47 48 49  \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.  Ralf Jung committed Mar 11, 2016 50 51  \clearpage  Ralf Jung committed Mar 22, 2016 52 \section{RA and CMRA constructions}  Ralf Jung committed Feb 29, 2016 53   Ralf Jung committed Mar 11, 2016 54 55 56 \subsection{Product} \label{sec:prodm}  Ralf Jung committed Mar 11, 2016 57 Given a family $(M_i)_{i \in I}$ of CMRAs ($I$ finite), we construct a CMRA for the product $\prod_{i \in I} M_i$ by lifting everything pointwise.  Ralf Jung committed Mar 11, 2016 58 59 60 61 62  Frame-preserving updates on the $M_i$ lift to the product: \begin{mathpar} \inferH{prod-update} {\melt \mupd_{M_i} \meltsB}  Robbert Krebbers committed Oct 17, 2016 63  {\mapinsert i \melt f \mupd \setComp{ \mapinsert i \meltB f}{\meltB \in \meltsB}}  Ralf Jung committed Mar 11, 2016 64 65 \end{mathpar}  Ralf Jung committed Jul 25, 2016 66 67 68 \subsection{Sum} \label{sec:summ}  Ralf Jung committed Jul 27, 2016 69 The \emph{sum CMRA} $\monoid_1 \csumm \monoid_2$ for any CMRAs $\monoid_1$ and $\monoid_2$ is defined as (again, we use a datatype-like notation):  Ralf Jung committed Jul 25, 2016 70 71 72 73 74 75 76 77 78 79 80 81 82 \begin{align*} \monoid_1 \csumm \monoid_2 \eqdef{}& \cinl(\melt_1:\monoid_1) \mid \cinr(\melt_2:\monoid_2) \mid \bot \\ \mval_n \eqdef{}& \setComp{\cinl(\melt_1)\!}{\!\melt_1 \in \mval'_n} \cup \setComp{\cinr(\melt_2)\!}{\!\melt_2 \in \mval''_n} \\ \cinl(\melt_1) \mtimes \cinl(\meltB_1) \eqdef{}& \cinl(\melt_1 \mtimes \meltB_1) \\ % \munit \mtimes \ospending \eqdef{}& \ospending \mtimes \munit \eqdef \ospending \\ % \munit \mtimes \osshot(\melt) \eqdef{}& \osshot(\melt) \mtimes \munit \eqdef \osshot(\melt) \\ \mcore{\cinl(\melt_1)} \eqdef{}& \begin{cases}\mnocore & \text{if $\mcore{\melt_1} = \mnocore$} \\ \cinl({\mcore{\melt_1}}) & \text{otherwise} \end{cases} \end{align*} The composition and core for $\cinr$ are defined symmetrically. The remaining cases of the composition and core are all $\bot$. Above, $\mval'$ refers to the validity of $\monoid_1$, and $\mval''$ to the validity of $\monoid_2$.  Ralf Jung committed Oct 18, 2016 83 84 85 86 87 88 89 90 91 92 The step-indexed equivalence is inductively defined as follows: \begin{mathpar} \infer{x \nequiv{n} y}{\cinl(x) \nequiv{n} \cinl(y)} \infer{x \nequiv{n} y}{\cinr(x) \nequiv{n} \cinr(y)} \axiom{\bot \nequiv{n} \bot} \end{mathpar}  Ralf Jung committed Jul 25, 2016 93 94 95 96 97 98 99 100 101 102 103 104 We obtain the following frame-preserving updates, as well as their symmetric counterparts: \begin{mathpar} \inferH{sum-update} {\melt \mupd_{M_1} \meltsB} {\cinl(\melt) \mupd \setComp{ \cinl(\meltB)}{\meltB \in \meltsB}} \inferH{sum-swap} {\All \melt_\f, n. \melt \mtimes \melt_\f \notin \mval'_n \and \meltB \in \mval''} {\cinl(\melt) \mupd \cinr(\meltB)} \end{mathpar} Crucially, the second rule allows us to \emph{swap} the side'' of the sum that the CMRA is on if $\mval$ has \emph{no possible frame}.  Ralf Jung committed Oct 18, 2016 105 106 107 108 109 110 111 112 113 114 \subsection{Option} The definition of the (CM)RA axioms already lifted the composition operation on $\monoid$ to one on $\maybe\monoid$. We can easily extend this to a full CMRA by defining a suitable core, namely \begin{align*} \mcore{\mnocore} \eqdef{}& \mnocore & \\ \mcore{\maybe\melt} \eqdef{}& \mcore\melt & \text{If $\maybe\melt \neq \mnocore$} \end{align*} Notice that this core is total, as the result always lies in $\maybe\monoid$ (rather than in $\maybe{\maybe\monoid}$).  Ralf Jung committed Mar 11, 2016 115 116 117 \subsection{Finite partial function} \label{sec:fpfnm}  Ralf Jung committed Aug 11, 2016 118 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.  Ralf Jung committed Mar 11, 2016 119 120 121 122 123  We obtain the following frame-preserving updates: \begin{mathpar} \inferH{fpfn-alloc-strong} {\text{$G$ infinite} \and \melt \in \mval}  Robbert Krebbers committed Oct 17, 2016 124  {\emptyset \mupd \setComp{\mapsingleton \gname \melt}{\gname \in G}}  Ralf Jung committed Mar 11, 2016 125 126 127  \inferH{fpfn-alloc} {\melt \in \mval}  Robbert Krebbers committed Oct 17, 2016 128  {\emptyset \mupd \setComp{\mapsingleton \gname \melt}{\gname \in K}}  Ralf Jung committed Mar 11, 2016 129 130  \inferH{fpfn-update}  Ralf Jung committed Jul 27, 2016 131  {\melt \mupd_\monoid \meltsB}  Robbert Krebbers committed Oct 17, 2016 132  {\mapinsert i \melt f] \mupd \setComp{ \mapinsert i \meltB f}{\meltB \in \meltsB}}  Ralf Jung committed Mar 11, 2016 133 \end{mathpar}  Ralf Jung committed Jul 27, 2016 134 Above, $\mval$ refers to the validity of $\monoid$.  Ralf Jung committed Mar 23, 2016 135   Ralf Jung committed Mar 11, 2016 136 $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.  Ralf Jung committed Mar 11, 2016 137   Ralf Jung committed Feb 29, 2016 138 139 \subsection{Agreement}  Ralf Jung committed Mar 09, 2016 140 141 Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows: \begin{align*}  Robbert Krebbers committed Oct 17, 2016 142  \agm(\cofe) \eqdef{}& \set{(c, V) \in (\nat \to \cofe) \times \SProp}/\ {\sim} \-0.2em]  Ralf Jung committed Jul 27, 2016 143 144 145 146 147  \textnormal{where }& \melt \sim \meltB \eqdef{} \melt.V = \meltB.V \land \All n. n \in \melt.V \Ra \melt.c(n) \nequiv{n} \meltB.c(n) \\ % \All n \in {\melt.V}.\, \melt.x \nequiv{n} \meltB.x \\ \melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.V \Lra m \in \meltB.V) \land (\All m \leq n. m \in \melt.V \Ra \melt.c(m) \nequiv{m} \meltB.c(m)) \\ \mval_n \eqdef{}& \setComp{\melt \in \agm(\cofe)}{ n \in \melt.V \land \All m \leq n. \melt.c(n) \nequiv{m} \melt.c(m) } \\  Ralf Jung committed Mar 09, 2016 148  \mcore\melt \eqdef{}& \melt \\  Ralf Jung committed Jul 27, 2016 149  \melt \mtimes \meltB \eqdef{}& \left(\melt.c, \setComp{n}{n \in \melt.V \land n \in \meltB.V \land \melt \nequiv{n} \meltB }\right)  Ralf Jung committed Mar 09, 2016 150 \end{align*}  Ralf Jung committed Jul 27, 2016 151 %Note that the carrier \agm(\cofe) is a \emph{record} consisting of the two fields c and V.  Ralf Jung committed Mar 23, 2016 152   Ralf Jung committed Mar 11, 2016 153 \agm(-) is a locally non-expansive functor from \COFEs to \CMRAs.  Ralf Jung committed Mar 09, 2016 154   Ralf Jung committed Jul 27, 2016 155 You can think of the c as a \emph{chain} of elements of \cofe that has to converge only for n \in V steps.  Ralf Jung committed Mar 11, 2016 156 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).  Ralf Jung committed Jul 27, 2016 157 However, given such a chain, we cannot constructively define its limit: Clearly, the V of the limit is the limit of the V of the chain.  Ralf Jung committed Mar 11, 2016 158 But what to pick for the actual data, for the element of \cofe?  Robbert Krebbers committed Oct 17, 2016 159 Only if V = \nat we have a chain of \cofe that we can take a limit of; if the V is smaller, the chain cancels'', \ie stops converging as we reach indices n \notin V.  Ralf Jung committed Mar 11, 2016 160 To mitigate this, we apply the usual construction to close a set; we go from elements of \cofe to chains of \cofe.  Ralf Jung committed Mar 09, 2016 161   Ralf Jung committed Mar 12, 2016 162 We define an injection \aginj into \agm(\cofe) as follows:  Robbert Krebbers committed Oct 17, 2016 163 \[ \aginj(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \nat}  Ralf Jung committed Mar 09, 2016 164 165 There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following: \begin{mathpar}  Ralf Jung committed Mar 12, 2016 166  \axiomH{ag-val}{\aginj(x) \in \mval_n}  Ralf Jung committed Mar 11, 2016 167   Ralf Jung committed Mar 12, 2016 168  \axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)}  Ralf Jung committed Mar 11, 2016 169   Ralf Jung committed Mar 12, 2016 170  \axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Ra x \nequiv{n} y}  Ralf Jung committed Mar 09, 2016 171 172 \end{mathpar}  Ralf Jung committed Feb 29, 2016 173   Ralf Jung committed Mar 12, 2016 174 175 \subsection{Exclusive CMRA}  Ralf Jung committed Aug 11, 2016 176 Given a COFE $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned:  Ralf Jung committed Mar 12, 2016 177 \begin{align*}  Ralf Jung committed Jul 25, 2016 178 179  \exm(\cofe) \eqdef{}& \exinj(\cofe) + \bot \\ \mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \bot}  Ralf Jung committed Mar 12, 2016 180 \end{align*}  Ralf Jung committed Jul 25, 2016 181 All cases of composition go to $\bot$.  Ralf Jung committed Mar 12, 2016 182 \begin{align*}  Ralf Jung committed Jul 25, 2016 183  \mcore{\exinj(x)} \eqdef{}& \mnocore &  Ralf Jung committed Mar 12, 2016 184 185  \mcore{\bot} \eqdef{}& \bot \end{align*}  Ralf Jung committed Jul 28, 2016 186 187 Remember that $\mnocore$ is the dummy'' element in $\maybe\monoid$ indicating (in this case) that $\exinj(x)$ has no core.  Ralf Jung committed Mar 12, 2016 188 189 190 The step-indexed equivalence is inductively defined as follows: \begin{mathpar} \infer{x \nequiv{n} y}{\exinj(x) \nequiv{n} \exinj(y)}  Ralf Jung committed Feb 29, 2016 191   Ralf Jung committed Mar 12, 2016 192 193 194 195 196 197 198 199 200 201 202 203 204  \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  Ralf Jung committed Feb 29, 2016 205 206 207 208 209 210 211 212 213 214 215 216 217 218 % \subsection{Finite Powerset Monoid} % Given an infinite set $X$, we define a monoid $\textmon{PowFin}$ with carrier $\mathcal{P}^{\textrm{fin}}(X)$ as follows: % $% \melt \cdot \meltB \;\eqdef\; \melt \cup \meltB \quad \mbox{if } \melt \cap \meltB = \emptyset %$ % We obtain: % \begin{mathpar} % \inferH{PowFinUpd}{} % {\emptyset \mupd \{ \{x\} \mid x \in X \}} % \end{mathpar} % \begin{proof}[Proof of \ruleref{PowFinUpd}]  Ralf Jung committed Mar 12, 2016 219 % Assume some frame $\melt_\f \sep \emptyset$. Since $\melt_\f$ is finite and $X$ is infinite, there exists an $x \notin \melt_\f$.  Ralf Jung committed Feb 29, 2016 220 221 222 223 224 % Pick that for the result. % \end{proof} % The powerset monoids is cancellative. % \begin{proof}[Proof of cancellativity]  Ralf Jung committed Mar 12, 2016 225 226 227 228 % Let $\melt_\f \mtimes \melt = \melt_\f \mtimes \meltB \neq \mzero$. % So we have $\melt_\f \sep \melt$ and $\melt_\f \sep \meltB$, and we have to show $\melt = \meltB$. % Assume $x \in \melt$. Hence $x \in \melt_\f \mtimes \melt$ and thus $x \in \melt_\f \mtimes \meltB$. % By disjointness, $x \notin \melt_\f$ and hence $x \in meltB$.  Ralf Jung committed Feb 29, 2016 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 % The other direction works the same way. % \end{proof} % \subsection{Fractional monoid} % \label{sec:fracm} % Given a monoid $M$, we define a monoid representing fractional ownership of some piece $\melt \in M$. % The idea is to preserve all the frame-preserving update that $M$ could have, while additionally being able to do \emph{any} update if we own the full state (as determined by the fraction being $1$). % Let $\fracm{M}$ be the monoid with carrier $(((0, 1] \cap \mathbb{Q}) \times M) \uplus \{\munit\}$ and multiplication % \begin{align*} % (q, a) \mtimes (q', a') &\eqdef (q + q', a \mtimes a') \qquad \mbox{if $q+q'\le 1$} \\ % (q, a) \mtimes \munit &\eqdef (q,a) \\ % \munit \mtimes (q,a) &\eqdef (q,a). % \end{align*} % We get the following frame-preserving update. % \begin{mathpar} % \inferH{FracUpdFull} % {a, b \in M} % {(1, a) \mupd (1, b)} % \and\inferH{FracUpdLocal} % {a \mupd_M B} % {(q, a) \mupd \{q\} \times B} % \end{mathpar} % \begin{proof}[Proof of \ruleref{FracUpdFull}] % Assume some $f \sep (1, a)$. This can only be $f = \munit$, so showing $f \sep (1, b)$ is trivial. % \end{proof} % \begin{proof}[Proof of \ruleref{FracUpdLocal}] % Assume some $f \sep (q, a)$. If $f = \munit$, then $f \sep (q, b)$ is trivial for any $b \in B$. Just pick the one we obtain by choosing $\munit_M$ as the frame for $a$.  Ralf Jung committed Jan 31, 2016 261   Ralf Jung committed Mar 12, 2016 262 263 % In the interesting case, we have $f = (q_\f, a_\f)$. % Obtain $b$ such that $b \in B \land b \sep a_\f$.  Ralf Jung committed Feb 29, 2016 264 265 266 267 268 % Then $(q, b) \sep f$, and we are done. % \end{proof} % $\fracm{M}$ is cancellative if $M$ is cancellative. % \begin{proof}[Proof of cancellativitiy]  Ralf Jung committed Mar 12, 2016 269 270 % If $\melt_\f = \munit$, we are trivially done. % So let $\melt_\f = (q_\f, \melt_\f')$.  Ralf Jung committed Feb 29, 2016 271 272 273 274 % If $\melt = \munit$, then $\meltB = \munit$ as otherwise the fractions could not match up. % Again, we are trivially done. % Similar so for $\meltB = \munit$. % So let $\melt = (q_a, \melt')$ and $\meltB = (q_b, \meltB')$.  Ralf Jung committed Mar 12, 2016 275 % We have $(q_\f + q_a, \melt_\f' \mtimes \melt') = (q_\f + q_b, \melt_\f' \mtimes \meltB')$.  Ralf Jung committed Feb 29, 2016 276 277 278 279 280 % We have to show $q_a = q_b$ and $\melt' = \meltB'$. % The first is trivial, the second follows from cancellativitiy of $M$. % \end{proof}  Ralf Jung committed Oct 15, 2016 281 282 \subsection{Authoritative} \label{sec:auth-cmra}  Ralf Jung committed Feb 29, 2016 283   Ralf Jung committed Oct 15, 2016 284 Given a CMRA $M$, we construct $\authm(M)$ modeling someone owning an \emph{authoritative} element $\melt$ of $M$, and others potentially owning fragments $\meltB \mincl \melt$ of $\melt$.  Ralf Jung committed Oct 15, 2016 285 286 287 288 289 290 291 292 293 294 We assume that $M$ has a unit $\munit$, and hence its core is total. (If $M$ is an exclusive monoid, the construction is very similar to a half-ownership monoid with two asymmetric halves.) \begin{align*} \authm(M) \eqdef{}& \maybe{\exm(M)} \times M \\ \mval_n \eqdef{}& \setComp{ (x, \meltB) \in \authm(M) }{ \meltB \in \mval_n \land (x = \mnocore \lor \Exists \melt. x = \exinj(\melt) \land \meltB \mincl_n \melt) } \\ (x_1, \meltB_1) \mtimes (x_2, \meltB_2) \eqdef{}& (x_1 \mtimes x_2, \meltB_2 \mtimes \meltB_2) \\ \mcore{(x, \meltB)} \eqdef{}& (\mnocore, \mcore\meltB) \\ (x_1, \meltB_1) \nequiv{n} (x_2, \meltB_2) \eqdef{}& x_1 \nequiv{n} x_2 \land \meltB_1 \nequiv{n} \meltB_2 \end{align*} Note that $(\mnocore, \munit)$ is the unit and asserts no ownership whatsoever, but $(\exinj(\munit), \munit)$ asserts that the authoritative element is $\munit$.  Ralf Jung committed Feb 29, 2016 295   Ralf Jung committed Oct 15, 2016 296 297 Let $\melt, \meltB \in M$. We write $\authfull \melt$ for full ownership $(\exinj(\melt), \munit)$ and $\authfrag \meltB$ for fragmental ownership $(\mnocore, \meltB)$ and $\authfull \melt , \authfrag \meltB$ for combined ownership $(\exinj(\melt), \meltB)$.  Ralf Jung committed Feb 29, 2016 298   Ralf Jung committed Oct 15, 2016 299 300 301 302 303 304 305 The frame-preserving update involves the notion of a \emph{local update}: \newcommand\lupd{\stackrel{\mathrm l}{\mupd}} \begin{defn} It is possible to do a \emph{local update} from $\melt_1$ and $\meltB_1$ to $\melt_2$ and $\meltB_2$, written $(\melt_1, \meltB_1) \lupd (\melt_2, \meltB_2)$, if $\All n, \maybe{\melt_\f}. x_1 \in \mval_n \land \melt_1 \nequiv{n} \meltB_1 \mtimes \maybe{\melt_\f} \Ra \melt_2 \in \mval_n \land \melt_2 \nequiv{n} \meltB_2 \mtimes \maybe{\melt_\f}$ \end{defn} In other words, the idea is that for every possible frame $\maybe{\melt_\f}$ completing $\meltB_1$ to $\melt_1$, the same frame also completes $\meltB_2$ to $\melt_2$.  Ralf Jung committed Feb 29, 2016 306   Ralf Jung committed Oct 15, 2016 307 308 309 310 311 312 We then obtain \begin{mathpar} \inferH{auth-update} {(\melt_1, \meltB_1) \lupd (\melt_2, \meltB_2)} {\authfull \melt_1 , \authfrag \meltB_1 \mupd \authfull \melt_2 , \authfrag \meltB_2} \end{mathpar}  Ralf Jung committed Feb 29, 2016 313   Ralf Jung committed Mar 22, 2016 314 \subsection{STS with tokens}  Ralf Jung committed Oct 15, 2016 315 \label{sec:sts-cmra}  Ralf Jung committed Feb 29, 2016 316   Ralf Jung committed Aug 11, 2016 317 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 an RA modeling an authoritative current state and permitting transitions given a \emph{bound} on the current state and a set of \emph{locally-owned} tokens.  Ralf Jung committed Feb 29, 2016 318   Ralf Jung committed Mar 22, 2016 319 320 321 322 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' \\  Ralf Jung committed Jul 03, 2016 323  s \stsfstep{T} s' \eqdef{}& \Exists T_1, T_2. T_1 \disj \STSL(s) \cup T \land (s, T_1) \stsstep (s', T_2)  Ralf Jung committed Mar 22, 2016 324 \end{align*}  Ralf Jung committed Feb 29, 2016 325   Ralf Jung committed Mar 22, 2016 326 327 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*}  Ralf Jung committed Aug 17, 2016 328 \STSclsd(S, T) \eqdef{}& \All s \in S. \STSL(s) \disj T \land \left(\All s'. s \stsfstep{T} s' \Ra s' \in S\right) \\  Ralf Jung committed Mar 22, 2016 329 330 \upclose(S, T) \eqdef{}& \setComp{ s' \in \STSS}{\Exists s \in S. s \stsftrans{T} s' } \end{align*}  Ralf Jung committed Feb 29, 2016 331   Ralf Jung committed Mar 22, 2016 332 333 The STS RA is defined as follows \begin{align*}  Ralf Jung committed Mar 22, 2016 334 335 336  \monoid \eqdef{}& \setComp{\STSauth((s, T) \in \STSS \times \wp(\STST))}{\STSL(s) \disj T} +{}\\& \setComp{\STSfrag((S, T) \in \wp(\STSS) \times \wp(\STST))}{\STSclsd(S, T) \land S \neq \emptyset} + \bot \\ \STSfrag(S_1, T_1) \mtimes \STSfrag(S_2, T_2) \eqdef{}& \STSfrag(S_1 \cap S_2, T_1 \cup T_2) \qquad\qquad\qquad \text{if $T_1 \disj T_2$ and $S_1 \cap S_2 \neq \emptyset$} \\ \STSfrag(S, T) \mtimes \STSauth(s, T') \eqdef{}& \STSauth(s, T') \mtimes \STSfrag(S, T) \eqdef \STSauth(s, T \cup T') \qquad \text{if $T \disj T'$ and $s \in S$} \\  Ralf Jung committed Mar 22, 2016 337 338 339 340  \mcore{\STSfrag(S, T)} \eqdef{}& \STSfrag(\upclose(S, \emptyset), \emptyset) \\ \mcore{\STSauth(s, T)} \eqdef{}& \STSfrag(\upclose(\set{s}, \emptyset), \emptyset) \end{align*} The remaining cases are all $\bot$.  Ralf Jung committed Feb 29, 2016 341   Ralf Jung committed Mar 22, 2016 342 343 344 345 We will need the following frame-preserving update: \begin{mathpar} \inferH{sts-step}{(s, T) \ststrans (s', T')} {\STSauth(s, T) \mupd \STSauth(s', T')}  Ralf Jung committed Feb 29, 2016 346   Ralf Jung committed Mar 22, 2016 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378  \inferH{sts-weaken} {\STSclsd(S_2, T_2) \and S_1 \subseteq S_2 \and T_2 \subseteq T_1} {\STSfrag(S_1, T_1) \mupd \STSfrag(S_2, T_2)} \end{mathpar} \paragraph{The core is not a homomorphism.} 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: $\mcore\melt \mtimes \mcore\meltB = \mcore{\melt\mtimes\meltB}$ To see why, consider the following STS: \newcommand\st{\textlog{s}} \newcommand\tok{\textmon{t}} \begin{center} \begin{tikzpicture}[sts] \node at (0,0) (s1) {$\st_1$}; \node at (3,0) (s2) {$\st_2$}; \node at (9,0) (s3) {$\st_3$}; \node at (6,0) (s4) {$\st_4$\\$[\tok_1, \tok_2]$}; \path[sts_arrows] (s2) edge (s4); \path[sts_arrows] (s3) edge (s4); \end{tikzpicture} \end{center} Now consider the following two elements of the STS RA: $\melt \eqdef \STSfrag(\set{\st_1,\st_2}, \set{\tok_1}) \qquad\qquad \meltB \eqdef \STSfrag(\set{\st_1,\st_3}, \set{\tok_2})$ We have: \begin{mathpar} {\melt\mtimes\meltB = \STSfrag(\set{\st_1}, \set{\tok_1, \tok_2})}  Ralf Jung committed Jan 31, 2016 379   Ralf Jung committed Mar 22, 2016 380 381 382 383 384 385 386  {\mcore\melt = \STSfrag(\set{\st_1, \st_2, \st_4}, \emptyset)} {\mcore\meltB = \STSfrag(\set{\st_1, \st_3, \st_4}, \emptyset)} {\mcore\melt \mtimes \mcore\meltB = \STSfrag(\set{\st_1, \st_4}, \emptyset) \neq \mcore{\melt \mtimes \meltB} = \STSfrag(\set{\st_1}, \emptyset)} \end{mathpar}  Ralf Jung committed Jan 31, 2016 387 388 389 390 391  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: