constructions.tex 20.1 KB
 Ralf Jung committed Jan 31, 2016 1 % !TEX root = ./appendix.tex  Ralf Jung committed Mar 11, 2016 2 \section{COFE constructions}  Ralf Jung committed Jan 31, 2016 3   Ralf Jung committed Mar 11, 2016 4 5 6 7 8 9 10 \subsection{Next (type-level later)} Given a COFE $\cofe$, we define $\latert\cofe$ as follows: \begin{align*} \latert\cofe \eqdef{}& \latertinj(\cofe) \\ \latertinj(x) \nequiv{n} \latertinj(y) \eqdef{}& n = 0 \lor x \nequiv{n-1} y \end{align*}  Ralf Jung committed Mar 11, 2016 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 $\latert(-)$ is a locally \emph{contractive} functor from $\COFEs$ to $\COFEs$. \subsection{Uniform Predicates} Given a CMRA $\monoid$, we define the COFE $\UPred(\monoid)$ of \emph{uniform predicates} over $\monoid$ as follows: \begin{align*} \UPred(\monoid) \eqdef{} \setComp{\pred: \mathbb{N} \times \monoid \to \mProp}{ \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 12, 2016 28 We start by defining the COFE of \emph{step-indexed propositions}: For every step-index, we proposition either holds or does not hold.  Ralf Jung committed Mar 11, 2016 29 30 \begin{align*} \SProp \eqdef{}& \psetdown{\mathbb{N}} \\  Ralf Jung committed Mar 12, 2016 31  \eqdef{}& \setComp{\prop \in \pset{\mathbb{N}}}{ \All n, m. n \geq m \Ra n \in \prop \Ra m \in \prop } \\  Ralf Jung committed Mar 11, 2016 32 33 34 35  \prop \nequiv{n} \propB \eqdef{}& \All m \leq n. m \in \prop \Lra m \in \propB \end{align*} Now we can rewrite $\UPred(\monoid)$ as monotone step-indexed predicates over $\monoid$, where the definition of a monotone'' function here is a little funny. \begin{align*}  Ralf Jung committed Mar 12, 2016 36  \UPred(\monoid) \cong{}& \monoid \monra \SProp \\  Ralf Jung committed Mar 11, 2016 37 38 39  \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 40 41  \clearpage  Ralf Jung committed Feb 29, 2016 42 43 \section{CMRA constructions}  Ralf Jung committed Mar 11, 2016 44 45 46 \subsection{Product} \label{sec:prodm}  Ralf Jung committed Mar 11, 2016 47 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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68  Frame-preserving updates on the $M_i$ lift to the product: \begin{mathpar} \inferH{prod-update} {\melt \mupd_{M_i} \meltsB} {f[i \mapsto \melt] \mupd \setComp{ f[i \mapsto \meltB]}{\meltB \in \meltsB}} \end{mathpar} \subsection{Finite partial function} \label{sec:fpfnm} Given some 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. We obtain the following frame-preserving updates: \begin{mathpar} \inferH{fpfn-alloc-strong} {\text{$G$ infinite} \and \melt \in \mval} {\emptyset \mupd \setComp{[\gname \mapsto \melt]}{\gname \in G}} \inferH{fpfn-alloc} {\melt \in \mval}  Ralf Jung committed Mar 11, 2016 69  {\emptyset \mupd \setComp{[\gname \mapsto \melt]}{\gname \in K}}  Ralf Jung committed Mar 11, 2016 70 71 72 73 74  \inferH{fpfn-update} {\melt \mupd \meltsB} {f[i \mapsto \melt] \mupd \setComp{ f[i \mapsto \meltB]}{\meltB \in \meltsB}} \end{mathpar}  Ralf Jung committed Mar 11, 2016 75 $K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.  Ralf Jung committed Mar 11, 2016 76   Ralf Jung committed Feb 29, 2016 77 78 \subsection{Agreement}  Ralf Jung committed Mar 09, 2016 79 Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:  Ralf Jung committed Mar 12, 2016 80 81 \newcommand{\aginjc}{\mathrm{c}} % the "c" field of an agreement element \newcommand{\aginjV}{\mathrm{V}} % the "V" field of an agreement element  Ralf Jung committed Mar 09, 2016 82 \begin{align*}  Ralf Jung committed Mar 12, 2016 83  \agm(\cofe) \eqdef{}& \record{\aginjc : \mathbb{N} \to \cofe , \aginjV : \SProp} \\  Ralf Jung committed Mar 09, 2016 84  & \text{quotiented by} \\  Ralf Jung committed Mar 12, 2016 85 86 87  \melt \equiv \meltB \eqdef{}& \melt.\aginjV = \meltB.\aginjV \land \All n. n \in \melt.\aginjV \Ra \melt.\aginjc(n) \nequiv{n} \meltB.\aginjc(n) \\ \melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\aginjV \Lra m \in \meltB.\aginjV) \land (\All m \leq n. m \in \melt.\aginjV \Ra \melt.\aginjc(m) \nequiv{m} \meltB.\aginjc(m)) \\ \mval_n \eqdef{}& \setComp{\melt \in \monoid}{ n \in \melt.\aginjV \land \All m \leq n. \melt.\aginjc(n) \nequiv{m} \melt.\aginjc(m) } \\  Ralf Jung committed Mar 09, 2016 88  \mcore\melt \eqdef{}& \melt \\  Ralf Jung committed Mar 12, 2016 89  \melt \mtimes \meltB \eqdef{}& (\melt.\aginjc, \setComp{n}{n \in \melt.\aginjV \land n \in \meltB.\aginjV \land \melt \nequiv{n} \meltB })  Ralf Jung committed Mar 09, 2016 90 \end{align*}  Ralf Jung committed Mar 11, 2016 91 $\agm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$.  Ralf Jung committed Mar 09, 2016 92   Ralf Jung committed Mar 12, 2016 93 You can think of the $\aginjc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \aginjV$ steps.  Ralf Jung committed Mar 11, 2016 94 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 Mar 12, 2016 95 However, given such a chain, we cannot constructively define its limit: Clearly, the $\aginjV$ of the limit is the limit of the $\aginjV$ of the chain.  Ralf Jung committed Mar 11, 2016 96 But what to pick for the actual data, for the element of $\cofe$?  Ralf Jung committed Mar 12, 2016 97 Only if $\aginjV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\aginjV$ is smaller, the chain cancels'', \ie stops converging as we reach indices $n \notin \aginjV$.  Ralf Jung committed Mar 11, 2016 98 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 99   Ralf Jung committed Mar 12, 2016 100 101 We define an injection $\aginj$ into $\agm(\cofe)$ as follows: $\aginj(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}}$  Ralf Jung committed Mar 09, 2016 102 103 There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following: \begin{mathpar}  Ralf Jung committed Mar 12, 2016 104  \axiomH{ag-val}{\aginj(x) \in \mval_n}  Ralf Jung committed Mar 11, 2016 105   Ralf Jung committed Mar 12, 2016 106  \axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)}  Ralf Jung committed Mar 11, 2016 107   Ralf Jung committed Mar 12, 2016 108  \axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Ra x \nequiv{n} y}  Ralf Jung committed Mar 09, 2016 109 110 \end{mathpar}  Ralf Jung committed Mar 11, 2016 111 112 113 114 115 \subsection{One-shot} The purpose of the one-shot CMRA is to lazily initialize the state of a ghost location. Given some CMRA $\monoid$, we define $\oneshotm(\monoid)$ as follows: \begin{align*}  Ralf Jung committed Mar 11, 2016 116  \oneshotm(\monoid) \eqdef{}& \ospending + \osshot(\monoid) + \munit + \bot \\  Ralf Jung committed Mar 11, 2016 117  \mval_n \eqdef{}& \set{\ospending, \munit} \cup \setComp{\osshot(\melt)}{\melt \in \mval_n}  Ralf Jung committed Mar 12, 2016 118 119 \\%\end{align*} %\begin{align*}  Ralf Jung committed Mar 11, 2016 120 121 122  \osshot(\melt) \mtimes \osshot(\meltB) \eqdef{}& \osshot(\melt \mtimes \meltB) \\ \munit \mtimes \ospending \eqdef{}& \ospending \mtimes \munit \eqdef \ospending \\ \munit \mtimes \osshot(\melt) \eqdef{}& \osshot(\melt) \mtimes \munit \eqdef \osshot(\melt)  Ralf Jung committed Mar 12, 2016 123 \end{align*}%  Ralf Jung committed Mar 11, 2016 124 The remaining cases of composition go to $\bot$.  Ralf Jung committed Mar 12, 2016 125 126 127 128 \begin{align*} \mcore{\ospending} \eqdef{}& \munit & \mcore{\osshot(\melt)} \eqdef{}& \mcore\melt \\ \mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot \end{align*}  Ralf Jung committed Mar 11, 2016 129 130 131 132 133 134 135 136 137 138 The step-indexed equivalence is inductively defined as follows: \begin{mathpar} \axiom{\ospending \nequiv{n} \ospending} \infer{\melt \nequiv{n} \meltB}{\osshot(\melt) \nequiv{n} \osshot(\meltB)} \axiom{\munit \nequiv{n} \munit} \axiom{\bot \nequiv{n} \bot} \end{mathpar}  Ralf Jung committed Mar 11, 2016 139 $\oneshotm(-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.  Ralf Jung committed Mar 11, 2016 140   Ralf Jung committed Mar 11, 2016 141 142 143 144 145 146 147 148 149 150 We obtain the following frame-preserving updates: \begin{mathpar} \inferH{oneshot-shoot} {\melt \in \mval} {\ospending \mupd \osshot(\melt)} \inferH{oneshot-update} {\melt \mupd \meltsB} {\osshot(\melt) \mupd \setComp{\osshot(\meltB)}{\meltB \in \meltsB}} \end{mathpar}  Ralf Jung committed Feb 29, 2016 151   Ralf Jung committed Mar 12, 2016 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 \subsection{Exclusive CMRA} Given a cofe $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned: \begin{align*} \exm(\cofe) \eqdef{}& \exinj(\cofe) + \munit + \bot \\ \mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \bot} \\ \munit \mtimes \exinj(x) \eqdef{}& \exinj(x) \mtimes \munit \eqdef \exinj(x) \end{align*} The remaining cases of composition go to $\bot$. \begin{align*} \mcore{\exinj(x)} \eqdef{}& \munit & \mcore{\munit} \eqdef{}& \munit & \mcore{\bot} \eqdef{}& \bot \end{align*} The step-indexed equivalence is inductively defined as follows: \begin{mathpar} \infer{x \nequiv{n} y}{\exinj(x) \nequiv{n} \exinj(y)}  Ralf Jung committed Feb 29, 2016 168   Ralf Jung committed Mar 12, 2016 169  \axiom{\munit \nequiv{n} \munit}  Ralf Jung committed Feb 29, 2016 170   Ralf Jung committed Mar 12, 2016 171 172 173 174 175 176 177 178 179 180 181 182 183  \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 184 185 186 187 188 189 190 191 192 193 194 195 196 197 % \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 198 % 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 199 200 201 202 203 % Pick that for the result. % \end{proof} % The powerset monoids is cancellative. % \begin{proof}[Proof of cancellativity]  Ralf Jung committed Mar 12, 2016 204 205 206 207 % 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 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 % 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 240   Ralf Jung committed Mar 12, 2016 241 242 % 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 243 244 245 246 247 % 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 248 249 % If $\melt_\f = \munit$, we are trivially done. % So let $\melt_\f = (q_\f, \melt_\f')$.  Ralf Jung committed Feb 29, 2016 250 251 252 253 % 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 254 % We have $(q_\f + q_a, \melt_\f' \mtimes \melt') = (q_\f + q_b, \melt_\f' \mtimes \meltB')$.  Ralf Jung committed Feb 29, 2016 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 % We have to show $q_a = q_b$ and $\melt' = \meltB'$. % The first is trivial, the second follows from cancellativitiy of $M$. % \end{proof} % %\subsection{Disposable monoid} % % % %Given a monoid $M$, we construct a monoid where, having full ownership of an element $\melt$ of $M$, one can throw it away, transitioning to a dead element. % %Let \dispm{M} be the monoid with carrier $\mcarp{M} \uplus \{ \disposed \}$ and multiplication % %% The previous unit must remain the unit of the new monoid, as is is always duplicable and hence we could not transition to \disposed if it were not composable with \disposed % %\begin{align*} % % \melt \mtimes \meltB &\eqdef \melt \mtimes_M \meltB & \IF \melt \sep[M] \meltB \\ % % \disposed \mtimes \disposed &\eqdef \disposed \\ % % \munit_M \mtimes \disposed &\eqdef \disposed \mtimes \munit_M \eqdef \disposed % %\end{align*} % %The unit is the same as in $M$. % % % %The frame-preserving updates are % %\begin{mathpar} % % \inferH{DispUpd} % % {a \in \mcarp{M} \setminus \{\munit_M\} \and a \mupd_M B} % % {a \mupd B} % % \and % % \inferH{Dispose} % % {a \in \mcarp{M} \setminus \{\munit_M\} \and \All b \in \mcarp{M}. a \sep b \Ra b = \munit_M} % % {a \mupd \disposed} % %\end{mathpar} % % % %\begin{proof}[Proof of \ruleref{DispUpd}] % %Assume a frame $f$. If $f = \disposed$, then $a = \munit_M$, which is a contradiction. % %Thus $f \in \mcarp{M}$ and we can use $a \mupd_M B$. % %\end{proof} % % % %\begin{proof}[Proof of \ruleref{Dispose}] % %The second premiss says that $a$ has no non-trivial frame in $M$. To show the update, assume a frame $f$ in $\dispm{M}$. Like above, we get $f \in \mcarp{M}$, and thus $f = \munit_M$. But $\disposed \sep \munit_M$ is trivial, so we are done. % %\end{proof} % \subsection{Authoritative monoid}\label{sec:auth} % Given a monoid $M$, we construct a monoid modeling someone owning an \emph{authoritative} element $x$ of $M$, and others potentially owning fragments $\melt \le_M x$ of $x$. % (If $M$ is an exclusive monoid, the construction is very similar to a half-ownership monoid with two asymmetric halves.) % Let $\auth{M}$ be the monoid with carrier % $% \setComp{ (x, \melt) }{ x \in \mcarp{\exm{\mcarp{M}}} \land \melt \in \mcarp{M} \land (x = \munit_{\exm{\mcarp{M}}} \lor \melt \leq_M x) } %$ % and multiplication % $% (x, \melt) \mtimes (y, \meltB) \eqdef % (x \mtimes y, \melt \mtimes \meltB) \quad \mbox{if } x \sep y \land \melt \sep \meltB \land (x \mtimes y = \munit_{\exm{\mcarp{M}}} \lor \melt \mtimes \meltB \leq_M x \mtimes y) %$ % Note that $(\munit_{\exm{\mcarp{M}}}, \munit_M)$ is the unit and asserts no ownership whatsoever, but $(\munit_{M}, \munit_M)$ asserts that the authoritative element is $\munit_M$. % Let $x, \melt \in \mcarp M$. % We write $\authfull x$ for full ownership $(x, \munit_M):\auth{M}$ and $\authfrag \melt$ for fragmental ownership $(\munit_{\exm{\mcarp{M}}}, \melt)$ and $\authfull x , \authfrag \melt$ for combined ownership $(x, \melt)$. % If $x$ or $a$ is $\mzero_{M}$, then the sugar denotes $\mzero_{\auth{M}}$. % \ralf{This needs syncing with the Coq development.} % The frame-preserving update involves a rather unwieldy side-condition: % \begin{mathpar} % \inferH{AuthUpd}{  Ralf Jung committed Mar 12, 2016 315 % \All\melt_\f\in\mcar{\monoid}. \melt\sep\meltB \land \melt\mtimes\melt_\f \le \meltB\mtimes\melt_\f \Ra \melt'\mtimes\melt_\f \le \melt'\mtimes\meltB \and  Ralf Jung committed Feb 29, 2016 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 % \melt' \sep \meltB % }{ % \authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \melt' \mtimes \meltB, \authfrag \melt' % } % \end{mathpar} % We therefore derive two special cases. % \paragraph{Local frame-preserving updates.} % \newcommand\authupd{f}% % Following~\cite{scsl}, we say that $\authupd: \mcar{M} \ra \mcar{M}$ is \emph{local} if % $% \All a, b \in \mcar{M}. a \sep b \land \authupd(a) \neq \mzero \Ra \authupd(a \mtimes b) = \authupd(a) \mtimes b %$ % Then, % \begin{mathpar} % \inferH{AuthUpdLocal} % {\text{$\authupd$ local} \and \authupd(\melt)\sep\meltB} % {\authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \authupd(\melt) \mtimes \meltB, \authfrag \authupd(\melt)} % \end{mathpar} % \paragraph{Frame-preserving updates on cancellative monoids.} % Frame-preserving updates are also possible if we assume $M$ cancellative: % \begin{mathpar} % \inferH{AuthUpdCancel} % {\text{$M$ cancellative} \and \melt'\sep\meltB} % {\authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \melt' \mtimes \meltB, \authfrag \melt'} % \end{mathpar} % \subsection{Fractional heap monoid} % \label{sec:fheapm} % By combining the fractional, finite partial function, and authoritative monoids, we construct two flavors of heaps with fractional permissions and mention their important frame-preserving updates. % Hereinafter, we assume the set $\textdom{Val}$ of values is countable. % Given a set $Y$, define $\FHeap(Y) \eqdef \textdom{Val} \fpfn \fracm(Y)$ representing a fractional heap with codomain $Y$. % From \S\S\ref{sec:fracm} and~\ref{sec:fpfunm} we obtain the following frame-preserving updates as well as the fact that $\FHeap(Y)$ is cancellative. % \begin{mathpar} % \axiomH{FHeapUpd}{h[x \mapsto (1, y)] \mupd h[x \mapsto (1, y')]} \and % \axiomH{FHeapAlloc}{h \mupd \{\, h[x \mapsto (1, y)] \mid x \in \textdom{Val} \,\}} % \end{mathpar} % We will write $qh$ with $h : \textsort{Val} \fpfn Y$ for the function in $\FHeap(Y)$ mapping every $x \in \dom(h)$ to $(q, h(x))$, and everything else to $\munit$. % Define $\AFHeap(Y) \eqdef \auth{\FHeap(Y)}$ representing an authoritative fractional heap with codomain $Y$. % We easily obtain the following frame-preserving updates. % \begin{mathpar} % \axiomH{AFHeapUpd}{ % (\authfull h[x \mapsto (1, y)], \authfrag [x \mapsto (1, y)]) \mupd (\authfull h[x \mapsto (1, y')], \authfrag [x \mapsto (1, y')]) % } % \and % \inferH{AFHeapAdd}{ % x \notin \dom(h) % }{ % \authfull h \mupd (\authfull h[x \mapsto (q, y)], \authfrag [x \mapsto (q, y)]) % } % \and % \axiomH{AFHeapRemove}{ % (\authfull h[x \mapsto (q, y)], \authfrag [x \mapsto (q, y)]) \mupd \authfull h % } % \end{mathpar} % \subsection{STS with tokens monoid} % \label{sec:stsmon} % Given a state-transition system~(STS) $(\STSS, \ra)$, a set of tokens $\STSS$, and a labeling $\STSL: \STSS \ra \mathcal{P}(\STST)$ of \emph{protocol-owned} tokens for each state, we construct a monoid modeling an authoritative current state and permitting transitions given a \emph{bound} on the current state and a set of \emph{locally-owned} tokens. % The construction follows the idea of STSs as described in CaReSL \cite{caresl}. % We first lift the transition relation to $\STSS \times \mathcal{P}(\STST)$ (implementing a \emph{law of token conservation}) and define upwards closure: % \begin{align*} % (s, T) \ra (s', T') \eqdef&\, s \ra s' \land \STSL(s) \uplus T = \STSL(s') \uplus T' \\ % \textsf{frame}(s, T) \eqdef&\, (s, \STST \setminus (\STSL(s) \uplus T)) \\ % \upclose(S, T) \eqdef&\, \setComp{ s' \in \STSS}{\exists s \in S.\; \textsf{frame}(s, T) \ststrans \textsf{frame}(s', T) } % \end{align*} % \noindent % We have % \begin{quote} % If $(s, T) \ra (s', T')$\\  Ralf Jung committed Mar 12, 2016 395 396 % and $T_\f \sep (T \uplus \STSL(s))$,\\ % then $\textsf{frame}(s, T_\f) \ra \textsf{frame}(s', T_\f)$.  Ralf Jung committed Feb 29, 2016 397 398 % \end{quote} % \begin{proof}  Ralf Jung committed Mar 12, 2016 399 % This follows directly by framing the tokens in $\STST \setminus (T_\f \uplus T \uplus \STSL(s))$ around the given transition, which yields $(s, \STST \setminus (T_\f \uplus \STSL{T}(s))) \ra (s', T' \uplus (\STST \setminus (T_\f \uplus T \uplus \STSL{T}(s))))$.  Ralf Jung committed Feb 29, 2016 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 % This is exactly what we have to show, since we know $\STSL(s) \uplus T = \STSL(s') \uplus T'$. % \end{proof} % Let $\STSMon{\STSS}$ be the monoid with carrier % % \setComp{ (s, S, T) \in \exm{\STSS} \times \mathcal{P}(\STSS) \times \mathcal{P}(\STST) }{ \begin{aligned} &(s = \munit \lor s \in S) \land \upclose(S, T) = S \land{} \\& S \neq \emptyset \land \All s \in S. \STSL(s) \sep T \end{aligned} } % % and multiplication % % (s, S, T) \mtimes (s', S', T') \eqdef (s'' \eqdef s \mtimes_{\exm{\STSS}} s', S'' \eqdef S \cap S', T'' \eqdef T \cup T') \quad \text{if }\begin{aligned}[t] &(s = \munit \lor s' = \munit) \land T \sep T' \land{} \\& S'' \neq \emptyset \land (s'' \neq \munit \Ra s'' \in S'') \end{aligned} % % Some sugar makes it more convenient to assert being at least in a certain state and owning some tokens: $(s, T) : \STSMon{\STSS} \eqdef (\munit, \upclose(\{s\}, T), T) : \STSMon{\STSS}$, and % $s : \STSMon{\STSS} \eqdef (s, \emptyset) : \STSMon{\STSS}$. % We will need the following frame-preserving update. % \begin{mathpar} % \inferH{StsStep}{(s, T) \ststrans (s', T')} % {(s, S, T) \mupd (s', \upclose(\{s'\}, T'), T')} % \end{mathpar} % \begin{proof}[Proof of \ruleref{StsStep}]  Ralf Jung committed Mar 12, 2016 421 422 % Assume some upwards-closed $S_\f, T_\f$ (the frame cannot be authoritative) s.t.\ $s \in S_\f$ and $T_\f \sep (T \uplus \STSL(s))$. We have to show that this frame combines with our final monoid element, which is the case if $s' \in S_\f$ and $T_\f \sep T'$. % By upward-closedness, it suffices to show $\textsf{frame}(s, T_\f) \ststrans \textsf{frame}(s', T_\f)$.  Ralf Jung committed Feb 29, 2016 423 424 % This follows by induction on the path $(s, T) \ststrans (s', T')$, and using the lemma proven above for each step. % \end{proof}  Ralf Jung committed Jan 31, 2016 425   Ralf Jung committed Jan 31, 2016 426 427 428 429 430  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: