constructions.tex 17.9 KB
 Ralf Jung committed Dec 05, 2016 1 \section{OFE and COFE constructions}  Ralf Jung committed Jan 31, 2016 2   Ralf Jung committed Oct 18, 2016 3 4 \subsection{Trivial pointwise lifting}  Ralf Jung committed Dec 05, 2016 5 The (C)OFE structure on many types can be easily obtained by pointwise lifting of the structure of the components.  Ralf Jung committed Oct 18, 2016 6 7 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 Dec 05, 2016 10 Given a OFE $\cofe$, 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 Dec 05, 2016 17 $\latert(-)$ is a locally \emph{contractive} functor from $\OFEs$ to $\OFEs$.  Ralf Jung committed Mar 11, 2016 18   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 \begin{align*}  Ralf Jung committed Oct 18, 2016 71 72 73  \monoid_1 \csumm \monoid_2 \eqdef{}& \cinl(\melt_1:\monoid_1) \mid \cinr(\melt_2:\monoid_2) \mid \mundef \\ \mval_n \eqdef{}& \setComp{\cinl(\melt_1)}{\melt_1 \in \mval'_n} \cup \setComp{\cinr(\melt_2)}{\melt_2 \in \mval''_n} \\  Ralf Jung committed Jul 25, 2016 74 75 76 77 78 79  \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.  Ralf Jung committed Oct 18, 2016 80 The remaining cases of the composition and core are all $\mundef$.  Ralf Jung committed Jul 25, 2016 81 82 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 Notice that we added the artificial invalid'' (or undefined'') element $\mundef$ to this CMRA just in order to make certain compositions of elements (in this case, $\cinl$ and $\cinr$) invalid.  Ralf Jung committed Oct 18, 2016 85 86 87 88 89 90 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)}  Ralf Jung committed Oct 18, 2016 91  \axiom{\mundef \nequiv{n} \mundef}  Ralf Jung committed Oct 18, 2016 92 93 94 \end{mathpar}  Ralf Jung committed Jul 25, 2016 95 96 97 98 99 100 101 102 103 104 105 106 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 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*}  Ralf Jung committed Oct 18, 2016 115 Notice that this core is total, as the result always lies in $\maybe\monoid$ (rather than in $\maybe{\mathord{\maybe\monoid}}$).  Ralf Jung committed Oct 18, 2016 116   Ralf Jung committed Mar 11, 2016 117 118 119 \subsection{Finite partial function} \label{sec:fpfnm}  Ralf Jung committed Dec 05, 2016 120 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.  Ralf Jung committed Mar 11, 2016 121 122 123 124 125  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 126  {\emptyset \mupd \setComp{\mapsingleton \gname \melt}{\gname \in G}}  Ralf Jung committed Mar 11, 2016 127 128 129  \inferH{fpfn-alloc} {\melt \in \mval}  Robbert Krebbers committed Oct 17, 2016 130  {\emptyset \mupd \setComp{\mapsingleton \gname \melt}{\gname \in K}}  Ralf Jung committed Mar 11, 2016 131 132  \inferH{fpfn-update}  Ralf Jung committed Jul 27, 2016 133  {\melt \mupd_\monoid \meltsB}  Robbert Krebbers committed Oct 17, 2016 134  {\mapinsert i \melt f] \mupd \setComp{ \mapinsert i \meltB f}{\meltB \in \meltsB}}  Ralf Jung committed Mar 11, 2016 135 \end{mathpar}  Ralf Jung committed Jul 27, 2016 136 Above, $\mval$ refers to the validity of $\monoid$.  Ralf Jung committed Mar 23, 2016 137   Ralf Jung committed Mar 11, 2016 138 $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.  Ralf Jung committed Mar 11, 2016 139   Ralf Jung committed Feb 29, 2016 140 141 \subsection{Agreement}  Ralf Jung committed Dec 05, 2016 142 Given some OFE $\cofe$, we define the CMRA $\agm(\cofe)$ as follows:  Ralf Jung committed Mar 09, 2016 143 \begin{align*}  Ralf Jung committed Dec 05, 2016 144 145 146 147  \agm(\cofe) \eqdef{}& \setComp{\melt \in \finpset\cofe}{c \neq \emptyset} /\ {\sim} \-0.2em] \melt \nequiv{n} \meltB \eqdef{}& (\All x \in \melt. \Exists y \in \meltB. x \nequiv{n} y) \land (\All y \in \meltB. \Exists x \in \melt. x \nequiv{n} y) \\ \textnormal{where }& \melt \sim \meltB \eqdef{} \All n. \melt \nequiv{n} \meltB \\ ~\\  Ralf Jung committed Jul 27, 2016 148 % \All n \in {\melt.V}.\, \melt.x \nequiv{n} \meltB.x \\  Ralf Jung committed Dec 05, 2016 149  \mval_n \eqdef{}& \setComp{\melt \in \agm(\cofe)}{ \All x, y \in \melt. x \nequiv{n} y } \\  Ralf Jung committed Mar 09, 2016 150  \mcore\melt \eqdef{}& \melt \\  Ralf Jung committed Dec 05, 2016 151  \melt \mtimes \meltB \eqdef{}& \melt \cup \meltB  Ralf Jung committed Mar 09, 2016 152 \end{align*}  Ralf Jung committed Jul 27, 2016 153 %Note that the carrier \agm(\cofe) is a \emph{record} consisting of the two fields c and V.  Ralf Jung committed Mar 23, 2016 154   Ralf Jung committed Dec 05, 2016 155 \agm(-) is a locally non-expansive functor from \OFEs to \CMRAs.  Ralf Jung committed Mar 09, 2016 156   Ralf Jung committed Dec 05, 2016 157 158 We define a non-expansive injection \aginj into \agm(\cofe) as follows: \[ \aginj(x) \eqdef \set{x}  Ralf Jung committed Mar 09, 2016 159 160 There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following: \begin{mathpar}  Ralf Jung committed Mar 12, 2016 161  \axiomH{ag-val}{\aginj(x) \in \mval_n}  Ralf Jung committed Mar 11, 2016 162   Ralf Jung committed Mar 12, 2016 163  \axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)}  Ralf Jung committed Mar 11, 2016 164   Ralf Jung committed Dec 05, 2016 165  \axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Lra x \nequiv{n} y}  Ralf Jung committed Mar 09, 2016 166 167 \end{mathpar}  Ralf Jung committed Feb 29, 2016 168   Ralf Jung committed Mar 12, 2016 169 170 \subsection{Exclusive CMRA}  Ralf Jung committed Dec 05, 2016 171 Given an OFE $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned:  Ralf Jung committed Mar 12, 2016 172 \begin{align*}  Ralf Jung committed Oct 18, 2016 173 174  \exm(\cofe) \eqdef{}& \exinj(\cofe) \mid \mundef \\ \mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \mundef}  Ralf Jung committed Mar 12, 2016 175 \end{align*}  Ralf Jung committed Oct 18, 2016 176 All cases of composition go to $\mundef$.  Ralf Jung committed Mar 12, 2016 177 \begin{align*}  Ralf Jung committed Jul 25, 2016 178  \mcore{\exinj(x)} \eqdef{}& \mnocore &  Ralf Jung committed Oct 18, 2016 179  \mcore{\mundef} \eqdef{}& \mundef  Ralf Jung committed Mar 12, 2016 180 \end{align*}  Ralf Jung committed Jul 28, 2016 181 182 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 183 184 185 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 186   Ralf Jung committed Oct 18, 2016 187  \axiom{\mundef \nequiv{n} \mundef}  Ralf Jung committed Mar 12, 2016 188 \end{mathpar}  Ralf Jung committed Dec 05, 2016 189 $\exm(-)$ is a locally non-expansive functor from $\OFEs$ to $\CMRAs$.  Ralf Jung committed Mar 12, 2016 190 191 192 193 194 195 196 197 198 199  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 200 201 202 203 204 205 206 207 208 209 210 211 212 213 % \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 214 % 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 215 216 217 218 219 % Pick that for the result. % \end{proof} % The powerset monoids is cancellative. % \begin{proof}[Proof of cancellativity]  Ralf Jung committed Mar 12, 2016 220 221 222 223 % 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 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 % 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 256   Ralf Jung committed Mar 12, 2016 257 258 % 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 259 260 261 262 263 % 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 264 265 % If $\melt_\f = \munit$, we are trivially done. % So let $\melt_\f = (q_\f, \melt_\f')$.  Ralf Jung committed Feb 29, 2016 266 267 268 269 % 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 270 % We have $(q_\f + q_a, \melt_\f' \mtimes \melt') = (q_\f + q_b, \melt_\f' \mtimes \meltB')$.  Ralf Jung committed Feb 29, 2016 271 272 273 274 275 % 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 276 277 \subsection{Authoritative} \label{sec:auth-cmra}  Ralf Jung committed Feb 29, 2016 278   Ralf Jung committed Oct 15, 2016 279 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 280 281 282 283 284 285 286 287 288 289 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 290   Ralf Jung committed Oct 15, 2016 291 292 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 293   Ralf Jung committed Oct 15, 2016 294 295 296 297 298 299 300 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 301   Ralf Jung committed Oct 15, 2016 302 303 304 305 306 307 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 308   Ralf Jung committed Mar 22, 2016 309 \subsection{STS with tokens}  Ralf Jung committed Oct 15, 2016 310 \label{sec:sts-cmra}  Ralf Jung committed Feb 29, 2016 311   Ralf Jung committed Aug 11, 2016 312 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 313   Ralf Jung committed Mar 22, 2016 314 315 316 317 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 318  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 319 \end{align*}  Ralf Jung committed Feb 29, 2016 320   Ralf Jung committed Mar 22, 2016 321 322 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 323 \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 324 325 \upclose(S, T) \eqdef{}& \setComp{ s' \in \STSS}{\Exists s \in S. s \stsftrans{T} s' } \end{align*}  Ralf Jung committed Feb 29, 2016 326   Ralf Jung committed Mar 22, 2016 327 328 The STS RA is defined as follows \begin{align*}  Ralf Jung committed Oct 18, 2016 329 330  \monoid \eqdef{}& \STSauth(s:\STSS, T:\wp(\STST) \mid \STSL(s) \disj T) \mid{}\\& \STSfrag(S: \wp(\STSS), T: \wp(\STST) \mid \STSclsd(S, T) \land S \neq \emptyset) \mid \mundef \\ \mval \eqdef{}& \setComp{\melt\in\monoid}{\melt \neq \mundef} \\  Ralf Jung committed Mar 22, 2016 331 332  \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 333 334 335  \mcore{\STSfrag(S, T)} \eqdef{}& \STSfrag(\upclose(S, \emptyset), \emptyset) \\ \mcore{\STSauth(s, T)} \eqdef{}& \STSfrag(\upclose(\set{s}, \emptyset), \emptyset) \end{align*}  Ralf Jung committed Oct 18, 2016 336 The remaining cases are all $\mundef$.  Ralf Jung committed Feb 29, 2016 337   Ralf Jung committed Mar 22, 2016 338 339 340 341 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 342   Ralf Jung committed Mar 22, 2016 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374  \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 375   Ralf Jung committed Mar 22, 2016 376 377 378 379 380 381 382  {\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 383 384 385 386 387  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: