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   Robbert Krebbers committed Dec 10, 2017 20 \subsection{Uniform predicates}  Ralf Jung committed Mar 11, 2016 21 22 23  Given a CMRA $\monoid$, we define the COFE $\UPred(\monoid)$ of \emph{uniform predicates} over $\monoid$ as follows: \begin{align*}  Ralf Jung committed Dec 08, 2017 24 25 26 27 28 \monoid \monnra \SProp \eqdef{}& \setComp{\pred: \monoid \nfn \SProp} {\All n, \melt, \meltB. \melt \mincl[n] \meltB \Ra \pred(\melt) \nincl{n} \pred(\meltB)} \\ \UPred(\monoid) \eqdef{}& \faktor{\monoid \monnra \SProp}{\equiv} \\ \pred \equiv \predB \eqdef{}& \All m, \melt. m \in \mval(\melt) \Ra (m \in \pred(\melt) \iff m \in \predB(\melt)) \\ \pred \nequiv{n} \predB \eqdef{}& \All m \le n, \melt. m \in \mval(\melt) \Ra (m \in \pred(\melt) \iff m \in \predB(\melt))  Ralf Jung committed Mar 11, 2016 29 \end{align*}  Ralf Jung committed Dec 08, 2017 30 You can think of uniform predicates as monotone, step-indexed predicates over a CMRA that ignore'' invalid elements (as defined by the quotient).  Ralf Jung committed Mar 14, 2016 31   Ralf Jung committed Dec 08, 2017 32 $\UPred(-)$ is a locally non-expansive functor from $\CMRAs$ to $\COFEs$.  Ralf Jung committed Mar 11, 2016 33   Ralf Jung committed Dec 08, 2017 34 35 36 It is worth noting that the above quotient admits canonical representatives. More precisely, one can show that every equivalence class contains exactly one element $P_0$ such that:  Ralf Jung committed Dec 10, 2017 37 38 39 \begin{align*} \All n, \melt. (\mval(\melt) \nincl{n} P_0(\melt)) \Ra n \in P_0(\melt) \tagH{UPred-canonical} \end{align*}  Ralf Jung committed Dec 08, 2017 40 41 42 43 44 45 46 47 48 49 50 51 52 Intuitively, this says that $P_0$ trivially holds whenever the resource is invalid. Starting from any element $P$, one can find this canonical representative by choosing $P_0(\melt) := \setComp{n}{n \in \mval(\melt) \Ra n \in P(\melt)}$. Hence, as an alternative definition of $\UPred$, we could use the set of canonical representatives. This alternative definition would save us from using a quotient. However, the definitions of the various connectives would get more complicated, because we have to make sure they all verify \ruleref{UPred-canonical}, which sometimes requires some adjustments. We would moreover need to prove one more property for every logical connective.  Ralf Jung committed Mar 11, 2016 53 \clearpage  Ralf Jung committed Mar 22, 2016 54 \section{RA and CMRA constructions}  Ralf Jung committed Feb 29, 2016 55   Ralf Jung committed Mar 11, 2016 56 57 58 \subsection{Product} \label{sec:prodm}  Ralf Jung committed Mar 11, 2016 59 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 60 61 62 63 64  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 65  {\mapinsert i \melt f \mupd \setComp{ \mapinsert i \meltB f}{\meltB \in \meltsB}}  Ralf Jung committed Mar 11, 2016 66 67 \end{mathpar}  Ralf Jung committed Jul 25, 2016 68 69 70 \subsection{Sum} \label{sec:summ}  Ralf Jung committed Jul 27, 2016 71 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 72 \begin{align*}  Ralf Jung committed Oct 18, 2016 73  \monoid_1 \csumm \monoid_2 \eqdef{}& \cinl(\melt_1:\monoid_1) \mid \cinr(\melt_2:\monoid_2) \mid \mundef \\  Ralf Jung committed Dec 08, 2017 74 75  \mval(\mundef) \eqdef{}& \emptyset \\ \mval(\cinl(\melt)) \eqdef{}& \mval_1(\melt) \\  Ralf Jung committed Jul 25, 2016 76 77 78 79 80  \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*}  Ralf Jung committed Dec 08, 2017 81 82 Above, $\mval_1$ refers to the validity of $\monoid_1$. The validity, composition and core for $\cinr$ are defined symmetrically.  Ralf Jung committed Oct 18, 2016 83 The remaining cases of the composition and core are all $\mundef$.  Ralf Jung committed Jul 25, 2016 84   Ralf Jung committed Oct 18, 2016 85 86 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 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)}  Ralf Jung committed Oct 18, 2016 93  \axiom{\mundef \nequiv{n} \mundef}  Ralf Jung committed Oct 18, 2016 94 95 96 \end{mathpar}  Ralf Jung committed Jul 25, 2016 97 98 99 100 101 102 103 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}  Ralf Jung committed Dec 08, 2017 104  {\All \melt_\f \in M, n. n \notin \mval(\melt \mtimes \melt_\f) \and \mvalFull(\meltB)}  Ralf Jung committed Jul 25, 2016 105 106 107 108  {\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 109 110 111 112 113 114 115 116 \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 117 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 118   Ralf Jung committed Mar 11, 2016 119 120 121 \subsection{Finite partial function} \label{sec:fpfnm}  Ralf Jung committed Dec 05, 2016 122 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 123 124 125 126  We obtain the following frame-preserving updates: \begin{mathpar} \inferH{fpfn-alloc-strong}  Ralf Jung committed Dec 08, 2017 127  {\text{$G$ infinite} \and \mvalFull(\melt)}  Robbert Krebbers committed Oct 17, 2016 128  {\emptyset \mupd \setComp{\mapsingleton \gname \melt}{\gname \in G}}  Ralf Jung committed Mar 11, 2016 129 130  \inferH{fpfn-alloc}  Ralf Jung committed Dec 08, 2017 131  {\mvalFull(\melt)}  Robbert Krebbers committed Oct 17, 2016 132  {\emptyset \mupd \setComp{\mapsingleton \gname \melt}{\gname \in K}}  Ralf Jung committed Mar 11, 2016 133 134  \inferH{fpfn-update}  Ralf Jung committed Jul 27, 2016 135  {\melt \mupd_\monoid \meltsB}  Robbert Krebbers committed Oct 17, 2016 136  {\mapinsert i \melt f] \mupd \setComp{ \mapinsert i \meltB f}{\meltB \in \meltsB}}  Ralf Jung committed Mar 11, 2016 137 \end{mathpar}  Ralf Jung committed Dec 08, 2017 138 Above, $\mvalFull$ refers to the (full) validity of $\monoid$.  Ralf Jung committed Mar 23, 2016 139   Ralf Jung committed Mar 11, 2016 140 $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.  Ralf Jung committed Mar 11, 2016 141   Ralf Jung committed Feb 29, 2016 142 143 \subsection{Agreement}  Ralf Jung committed Dec 05, 2016 144 Given some OFE $\cofe$, we define the CMRA $\agm(\cofe)$ as follows:  Ralf Jung committed Mar 09, 2016 145 \begin{align*}  Ralf Jung committed Feb 02, 2017 146  \agm(\cofe) \eqdef{}& \setComp{\melt \in \finpset\cofe}{\melt \neq \emptyset} /\ {\sim} \-0.2em]  Ralf Jung committed Dec 05, 2016 147 148 149  \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 150 % \All n \in {\melt.V}.\, \melt.x \nequiv{n} \meltB.x \\  Ralf Jung committed Dec 08, 2017 151  \mval(\melt) \eqdef{}& \setComp{n}{ \All x, y \in \melt. x \nequiv{n} y } \\  Ralf Jung committed Mar 09, 2016 152  \mcore\melt \eqdef{}& \melt \\  Ralf Jung committed Dec 05, 2016 153  \melt \mtimes \meltB \eqdef{}& \melt \cup \meltB  Ralf Jung committed Mar 09, 2016 154 \end{align*}  Ralf Jung committed Jul 27, 2016 155 %Note that the carrier \agm(\cofe) is a \emph{record} consisting of the two fields c and V.  Ralf Jung committed Mar 23, 2016 156   Ralf Jung committed Dec 05, 2016 157 \agm(-) is a locally non-expansive functor from \OFEs to \CMRAs.  Ralf Jung committed Mar 09, 2016 158   Ralf Jung committed Dec 05, 2016 159 160 We define a non-expansive injection \aginj into \agm(\cofe) as follows: \[ \aginj(x) \eqdef \set{x}  Ralf Jung committed Mar 09, 2016 161 162 There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following: \begin{mathpar}  Ralf Jung committed Dec 08, 2017 163  \axiomH{ag-val}{\mvalFull(\aginj(x))}  Ralf Jung committed Mar 11, 2016 164   Ralf Jung committed Mar 12, 2016 165  \axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)}  Ralf Jung committed Mar 11, 2016 166   Ralf Jung committed Dec 08, 2017 167  \axiomH{ag-agree}{n \in \mval(\aginj(x) \mtimes \aginj(y)) \Ra x \nequiv{n} y}  Ralf Jung committed Mar 09, 2016 168 169 \end{mathpar}  Ralf Jung committed Feb 29, 2016 170   Ralf Jung committed Mar 12, 2016 171 172 \subsection{Exclusive CMRA}  Ralf Jung committed Dec 05, 2016 173 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 174 \begin{align*}  Ralf Jung committed Oct 18, 2016 175  \exm(\cofe) \eqdef{}& \exinj(\cofe) \mid \mundef \\  Ralf Jung committed Dec 08, 2017 176  \mval(\melt) \eqdef{}& \setComp{n}{\melt \neq \mundef}  Ralf Jung committed Mar 12, 2016 177 \end{align*}  Ralf Jung committed Oct 18, 2016 178 All cases of composition go to $\mundef$.  Ralf Jung committed Mar 12, 2016 179 \begin{align*}  Ralf Jung committed Jul 25, 2016 180  \mcore{\exinj(x)} \eqdef{}& \mnocore &  Ralf Jung committed Oct 18, 2016 181  \mcore{\mundef} \eqdef{}& \mundef  Ralf Jung committed Mar 12, 2016 182 \end{align*}  Ralf Jung committed Jul 28, 2016 183 184 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 185 186 187 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 188   Ralf Jung committed Oct 18, 2016 189  \axiom{\mundef \nequiv{n} \mundef}  Ralf Jung committed Mar 12, 2016 190 \end{mathpar}  Ralf Jung committed Dec 05, 2016 191 $\exm(-)$ is a locally non-expansive functor from $\OFEs$ to $\CMRAs$.  Ralf Jung committed Mar 12, 2016 192 193 194 195 196 197 198 199 200 201  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 202 203 204 205 206 207 208 209 210 211 212 213 214 215 % \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 216 % 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 217 218 219 220 221 % Pick that for the result. % \end{proof} % The powerset monoids is cancellative. % \begin{proof}[Proof of cancellativity]  Ralf Jung committed Mar 12, 2016 222 223 224 225 % 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 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 % 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 258   Ralf Jung committed Mar 12, 2016 259 260 % 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 261 262 263 264 265 % 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 266 267 % If $\melt_\f = \munit$, we are trivially done. % So let $\melt_\f = (q_\f, \melt_\f')$.  Ralf Jung committed Feb 29, 2016 268 269 270 271 % 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 272 % We have $(q_\f + q_a, \melt_\f' \mtimes \melt') = (q_\f + q_b, \melt_\f' \mtimes \meltB')$.  Ralf Jung committed Feb 29, 2016 273 274 275 276 277 % 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 278 279 \subsection{Authoritative} \label{sec:auth-cmra}  Ralf Jung committed Feb 29, 2016 280   Ralf Jung committed Oct 15, 2016 281 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 282 283 284 285 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 \\  Ralf Jung committed Dec 08, 2017 286 \mval( (x, \meltB ) ) \eqdef{}& \setComp{ n }{ n \in \mval(\meltB) \land (x = \mnocore \lor \Exists \melt. x = \exinj(\melt) \land \meltB \mincl_n \melt) } \\  Ralf Jung committed Oct 15, 2016 287 288 289 290 291  (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 292   Ralf Jung committed Oct 15, 2016 293 294 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 295   Ralf Jung committed Oct 15, 2016 296 297 298 299 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  Ralf Jung committed Dec 08, 2017 300  $\All n, \maybe{\melt_\f}. n \in \mval(\melt_1) \land \melt_1 \nequiv{n} \meltB_1 \mtimes \maybe{\melt_\f} \Ra n \in \mval(\melt_2) \land \melt_2 \nequiv{n} \meltB_2 \mtimes \maybe{\melt_\f}$  Ralf Jung committed Oct 15, 2016 301 302 \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 303   Ralf Jung committed Oct 15, 2016 304 305 306 307 308 309 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 310   Ralf Jung committed Mar 22, 2016 311 \subsection{STS with tokens}  Ralf Jung committed Oct 15, 2016 312 \label{sec:sts-cmra}  Ralf Jung committed Feb 29, 2016 313   Ralf Jung committed Aug 11, 2016 314 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 315   Ralf Jung committed Mar 22, 2016 316 317 318 319 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 320  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 321 \end{align*}  Ralf Jung committed Feb 29, 2016 322   Ralf Jung committed Mar 22, 2016 323 324 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 325 \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 326 327 \upclose(S, T) \eqdef{}& \setComp{ s' \in \STSS}{\Exists s \in S. s \stsftrans{T} s' } \end{align*}  Ralf Jung committed Feb 29, 2016 328   Ralf Jung committed Mar 22, 2016 329 330 The STS RA is defined as follows \begin{align*}  Ralf Jung committed Oct 18, 2016 331  \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 \\  Ralf Jung committed Dec 08, 2017 332  \mvalFull(\melt) \eqdef{}& \melt \neq \mundef \\  Ralf Jung committed Mar 22, 2016 333 334  \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 335 336 337  \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 338 The remaining cases are all $\mundef$.  Ralf Jung committed Feb 29, 2016 339   Ralf Jung committed Mar 22, 2016 340 341 342 343 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 344   Ralf Jung committed Mar 22, 2016 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376  \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 377   Ralf Jung committed Mar 22, 2016 378 379 380 381 382 383 384  {\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 385 386 387 388 389  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: