constructions.tex 16.7 KB
 Ralf Jung committed Dec 10, 2017 1 \section{OFE and COFE Constructions}  Ralf Jung committed Jan 31, 2016 2   Ralf Jung committed Dec 10, 2017 3 \subsection{Trivial Pointwise Lifting}  Ralf Jung committed Oct 18, 2016 4   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 Dec 10, 2017 8 \subsection{Next (Type-Level Later)}  Ralf Jung committed Mar 11, 2016 9   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 Dec 10, 2017 20 \subsection{Uniform Predicates}  Ralf Jung committed Mar 11, 2016 21   Robbert Krebbers committed Dec 10, 2017 22 Given a camera $\monoid$, we define the COFE $\UPred(\monoid)$ of \emph{uniform predicates} over $\monoid$ as follows:  Ralf Jung committed Mar 11, 2016 23 \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*}  Robbert Krebbers committed Dec 10, 2017 30 You can think of uniform predicates as monotone, step-indexed predicates over a camera 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  Robbert Krebbers committed Dec 10, 2017 54 \section{RA and Camera Constructions}  Ralf Jung committed Feb 29, 2016 55   Ralf Jung committed Mar 11, 2016 56 57 58 \subsection{Product} \label{sec:prodm}  Robbert Krebbers committed Dec 10, 2017 59 Given a family $(M_i)_{i \in I}$ of cameras ($I$ finite), we construct a camera 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}  Robbert Krebbers committed Dec 10, 2017 71 The \emph{sum camera} $\monoid_1 \csumm \monoid_2$ for any cameras $\monoid_1$ 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   Robbert Krebbers committed Dec 10, 2017 85 Notice that we added the artificial invalid'' (or undefined'') element $\mundef$ to this camera just in order to make certain compositions of elements (in this case, $\cinl$ and $\cinr$) invalid.  Ralf Jung committed Oct 18, 2016 86   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  {\cinl(\melt) \mupd \cinr(\meltB)} \end{mathpar}  Robbert Krebbers committed Dec 10, 2017 107 Crucially, the second rule allows us to \emph{swap} the side'' of the sum that the camera is on if $\mval$ has \emph{no possible frame}.  Ralf Jung committed Jul 25, 2016 108   Ralf Jung committed Oct 18, 2016 109 110 \subsection{Option}  Robbert Krebbers committed Dec 10, 2017 111 112 The definition of the camera/RA axioms already lifted the composition operation on $\monoid$ to one on $\maybe\monoid$. We can easily extend this to a full camera by defining a suitable core, namely  Ralf Jung committed Oct 18, 2016 113 114 115 116 \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 Dec 10, 2017 119 \subsection{Finite Partial Functions}  Ralf Jung committed Mar 11, 2016 120 121 \label{sec:fpfnm}  Robbert Krebbers committed Dec 10, 2017 122 Given some infinite countable $K$ and some camera $\monoid$, the set of finite partial functions $K \fpfn \monoid$ is equipped with a camera 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}  Robbert Krebbers committed Dec 10, 2017 144 Given some OFE $\cofe$, we define the camera $\agm(\cofe)$ 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   Robbert Krebbers committed Dec 10, 2017 171 \subsection{Exclusive Camera}  Ralf Jung committed Mar 12, 2016 172   Robbert Krebbers committed Dec 10, 2017 173 Given an OFE $\cofe$, we define a camera $\exm(\cofe)$ 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 11, 2017 176  \mval(\melt) \eqdef{}& \setComp{n}{\melt \notnequiv{n} \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  We obtain the following frame-preserving update: \begin{mathpar} \inferH{ex-update}{} {\exinj(x) \mupd \exinj(y)} \end{mathpar}  Ralf Jung committed Dec 11, 2017 199 \subsection{Fractions}  Ralf Jung committed Mar 12, 2016 200   Ralf Jung committed Dec 11, 2017 201 202 203 204 We define an RA structure on the rational numbers in $(0, 1]$ as follows: \begin{align*} \fracm \eqdef{}& \fracinj(\mathbb{Q} \cap (0, 1]) \mid \mundef \\ \mvalFull(\melt) \eqdef{}& \melt \neq \mundef \\  Ralf Jung committed Dec 11, 2017 205  \fracinj(q_1) \mtimes \fracinj(q_2) \eqdef{}& \fracinj(q_1 + q_2) \quad \text{if $q_1 + q_2 \leq 1$} \\  Ralf Jung committed Dec 11, 2017 206 207 208 209 210 211 212 213 214 215 216 217  \mcore{\fracinj(x)} \eqdef{}& \bot \\ \mcore{\mundef} \eqdef{}& \mundef \end{align*} All remaining cases of composition go to $\mundef$. Frequently, we will write just $x$ instead of $\fracinj(x)$. The most important property of this RA is that $1$ has no frame. This is useful in combination with \ruleref{sum-swap}, and also when used with pairs: \begin{mathpar} \inferH{pair-frac-change}{} {(1, a) \mupd (1, b)} \end{mathpar}  Ralf Jung committed Mar 12, 2016 218 219  %TODO: These need syncing with Coq  Ralf Jung committed Feb 29, 2016 220 221 222 223 224 225 226 227 228 229 230 231 232 233 % \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 234 % 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 235 236 237 238 239 % Pick that for the result. % \end{proof} % The powerset monoids is cancellative. % \begin{proof}[Proof of cancellativity]  Ralf Jung committed Mar 12, 2016 240 241 242 243 % 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 244 245 246 247 248 % The other direction works the same way. % \end{proof}  Ralf Jung committed Oct 15, 2016 249 \subsection{Authoritative}  Robbert Krebbers committed Dec 10, 2017 250 \label{sec:auth-camera}  Ralf Jung committed Feb 29, 2016 251   Robbert Krebbers committed Dec 10, 2017 252 Given a camera $M$, 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 253 254 255 256 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 Jun 29, 2018 257 \mval( (x, \meltB ) ) \eqdef{}& \setComp{ n }{ (x = \mnocore \land n \in \mval(\meltB)) \lor (\Exists \melt. x = \exinj(\melt) \land \meltB \mincl_n \melt \land n \in \mval(\melt)) } \\  Ralf Jung committed Oct 15, 2016 258 259 260 261 262  (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 263   Ralf Jung committed Oct 15, 2016 264 265 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 266   Ralf Jung committed Oct 15, 2016 267 268 269 270 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 271  $\All n, \maybe{\melt_\f}. n \in \mval(\melt_1) \land \melt_1 \nequiv{n} \meltB_1 \mtimes \maybe{\melt_\f} \Ra n \in \mval(\melt_2) \land \melt_2 \nequiv{n} \meltB_2 \mtimes \maybe{\melt_\f}$  Ralf Jung committed Oct 15, 2016 272 273 \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 274   Ralf Jung committed Oct 15, 2016 275 276 277 278 279 280 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 281   Ralf Jung committed Dec 10, 2017 282 \subsection{STS with Tokens}  Robbert Krebbers committed Dec 10, 2017 283 \label{sec:sts-camera}  Ralf Jung committed Feb 29, 2016 284   Ralf Jung committed Aug 11, 2016 285 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 286   Ralf Jung committed Mar 22, 2016 287 288 289 290 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 291  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 292 \end{align*}  Ralf Jung committed Feb 29, 2016 293   Ralf Jung committed Mar 22, 2016 294 295 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 296 \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 297 298 \upclose(S, T) \eqdef{}& \setComp{ s' \in \STSS}{\Exists s \in S. s \stsftrans{T} s' } \end{align*}  Ralf Jung committed Feb 29, 2016 299   Ralf Jung committed Mar 22, 2016 300 301 The STS RA is defined as follows \begin{align*}  Ralf Jung committed Oct 18, 2016 302  \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 303  \mvalFull(\melt) \eqdef{}& \melt \neq \mundef \\  Ralf Jung committed Mar 22, 2016 304 305  \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 306 307 308  \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 309 The remaining cases are all $\mundef$.  Ralf Jung committed Feb 29, 2016 310   Ralf Jung committed Mar 22, 2016 311 312 313 314 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 315   Ralf Jung committed Mar 22, 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  \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 348   Ralf Jung committed Mar 22, 2016 349 350 351 352 353 354 355  {\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 356 357 358 359 360  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: