program-logic.tex 14.6 KB
 Ralf Jung committed Oct 04, 2016 1 \section{Language}  Ralf Jung committed Oct 06, 2016 2 \label{sec:language}  Ralf Jung committed Oct 04, 2016 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61  A \emph{language} $\Lang$ consists of a set \textdom{Expr} of \emph{expressions} (metavariable $\expr$), a set \textdom{Val} of \emph{values} (metavariable $\val$), and a set \textdom{State} of \emph{states} (metvariable $\state$) such that \begin{itemize} \item There exist functions $\ofval : \textdom{Val} \to \textdom{Expr}$ and $\toval : \textdom{Expr} \pfn \textdom{val}$ (notice the latter is partial), such that \begin{mathpar} {\All \expr, \val. \toval(\expr) = \val \Ra \ofval(\val) = \expr} \and {\All\val. \toval(\ofval(\val)) = \val} \end{mathpar} \item There exists a \emph{primitive reduction relation} $(-,- \step -,-,-) \subseteq \textdom{Expr} \times \textdom{State} \times \textdom{Expr} \times \textdom{State} \times (\textdom{Expr} \uplus \set{\bot})$ We will write $\expr_1, \state_1 \step \expr_2, \state_2$ for $\expr_1, \state_1 \step \expr_2, \state_2, \bot$. \\ A reduction $\expr_1, \state_1 \step \expr_2, \state_2, \expr_\f$ indicates that, when $\expr_1$ reduces to $\expr_2$, a \emph{new thread} $\expr_\f$ is forked off. \item All values are stuck: $\expr, \_ \step \_, \_, \_ \Ra \toval(\expr) = \bot$ \end{itemize} \begin{defn} An expression $\expr$ and state $\state$ are \emph{reducible} (written $\red(\expr, \state)$) if $\Exists \expr_2, \state_2, \expr_\f. \expr,\state \step \expr_2,\state_2,\expr_\f$ \end{defn} \begin{defn} An expression $\expr$ is said to be \emph{atomic} if it reduces in one step to a value: $\All\state_1, \expr_2, \state_2, \expr_\f. \expr, \state_1 \step \expr_2, \state_2, \expr_\f \Ra \Exists \val_2. \toval(\expr_2) = \val_2$ \end{defn} \begin{defn}[Context] A function $\lctx : \textdom{Expr} \to \textdom{Expr}$ is a \emph{context} if the following conditions are satisfied: \begin{enumerate}[itemsep=0pt] \item $\lctx$ does not turn non-values into values:\\ $\All\expr. \toval(\expr) = \bot \Ra \toval(\lctx(\expr)) = \bot$ \item One can perform reductions below $\lctx$:\\ $\All \expr_1, \state_1, \expr_2, \state_2, \expr_\f. \expr_1, \state_1 \step \expr_2,\state_2,\expr_\f \Ra \lctx(\expr_1), \state_1 \step \lctx(\expr_2),\state_2,\expr_\f$ \item Reductions stay below $\lctx$ until there is a value in the hole:\\ $\All \expr_1', \state_1, \expr_2, \state_2, \expr_\f. \toval(\expr_1') = \bot \land \lctx(\expr_1'), \state_1 \step \expr_2,\state_2,\expr_\f \Ra \Exists\expr_2'. \expr_2 = \lctx(\expr_2') \land \expr_1', \state_1 \step \expr_2',\state_2,\expr_\f$ \end{enumerate} \end{defn} \subsection{Concurrent language} For any language $\Lang$, we define the corresponding thread-pool semantics. \paragraph{Machine syntax} $\tpool \in \textdom{ThreadPool} \eqdef \bigcup_n \textdom{Expr}^n$ \judgment[Machine reduction]{\cfg{\tpool}{\state} \step \cfg{\tpool'}{\state'}} \begin{mathpar} \infer {\expr_1, \state_1 \step \expr_2, \state_2, \expr_\f \and \expr_\f \neq \bot} {\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state_1} \step \cfg{\tpool \dplus [\expr_2] \dplus \tpool' \dplus [\expr_\f]}{\state_2}} \and\infer {\expr_1, \state_1 \step \expr_2, \state_2} {\cfg{\tpool \dplus [\expr_1] \dplus \tpool'}{\state_1} \step \cfg{\tpool \dplus [\expr_2] \dplus \tpool'}{\state_2}} \end{mathpar} \clearpage \section{Program Logic}  Ralf Jung committed Oct 06, 2016 62 \label{sec:program-logic}  Ralf Jung committed Oct 04, 2016 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244  \ralf{TODO: Right now, this is a dump of all the things that moved out of the base...} To instantiate Iris, you need to define the following parameters: \begin{itemize} \item A language $\Lang$, and \item a locally contractive bifunctor $\iFunc : \COFEs \to \CMRAs$ defining the ghost state, such that for all COFEs $\cofe$, the CMRA $\iFunc(A)$ has a unit. (By \lemref{lem:cmra-unit-total-core}, this means that the core of $\iFunc(\cofe)$ is a total function.) \end{itemize} We will write $\pvs[\term] \prop$ for $\pvs[\term][\term] \prop$. If we omit the mask, then it is $\top$ for weakest precondition $\wpre\expr{\Ret\var.\prop}$ and $\emptyset$ for primitive view shifts $\pvs \prop$. %FIXME $\top$ is not a term in the logic. Neither is any of the operations on masks that we use in the rules for weakestpre. Some propositions are \emph{timeless}, which intuitively means that step-indexing does not affect them. This is a \emph{meta-level} assertion about propositions, defined as follows: $\vctx \proves \timeless{\prop} \eqdef \vctx\mid\later\prop \proves \prop \lor \later\FALSE$ \paragraph{Metavariable conventions.} We introduce additional metavariables ranging over terms and generally let the choice of metavariable indicate the term's type: $\begin{array}{r|l} \text{metavariable} & \text{type} \\\hline \term, \termB & \text{arbitrary} \\ \val, \valB & \textlog{Val} \\ \expr & \textlog{Expr} \\ \state & \textlog{State} \\ \end{array} \qquad\qquad \begin{array}{r|l} \text{metavariable} & \text{type} \\\hline \iname & \textlog{InvName} \\ \mask & \textlog{InvMask} \\ \melt, \meltB & \textlog{M} \\ \prop, \propB, \propC & \Prop \\ \pred, \predB, \predC & \type\to\Prop \text{ (when \type is clear from context)} \\ \end{array}$ \begin{mathpar} \infer {\text{$\term$ or $\term'$ is a discrete COFE element}} {\timeless{\term =_\type \term'}} \infer {\text{$\melt$ is a discrete COFE element}} {\timeless{\ownGGhost\melt}} \infer {\text{$\melt$ is an element of a discrete CMRA}} {\timeless{\mval(\melt)}} \infer{} {\timeless{\ownPhys\state}} \infer {\vctx \proves \timeless{\propB}} {\vctx \proves \timeless{\prop \Ra \propB}} \infer {\vctx \proves \timeless{\propB}} {\vctx \proves \timeless{\prop \wand \propB}} \infer {\vctx,\var:\type \proves \timeless{\prop}} {\vctx \proves \timeless{\All\var:\type.\prop}} \infer {\vctx,\var:\type \proves \timeless{\prop}} {\vctx \proves \timeless{\Exists\var:\type.\prop}} \end{mathpar} \begin{mathpar} \infer[pvs-intro] {}{\prop \proves \pvs[\mask] \prop} \infer[pvs-mono] {\prop \proves \propB} {\pvs[\mask_1][\mask_2] \prop \proves \pvs[\mask_1][\mask_2] \propB} \infer[pvs-timeless] {\timeless\prop} {\later\prop \proves \pvs[\mask] \prop} \infer[pvs-trans] {\mask_2 \subseteq \mask_1 \cup \mask_3} {\pvs[\mask_1][\mask_2] \pvs[\mask_2][\mask_3] \prop \proves \pvs[\mask_1][\mask_3] \prop} \infer[pvs-mask-frame] {}{\pvs[\mask_1][\mask_2] \prop \proves \pvs[\mask_1 \uplus \mask_\f][\mask_2 \uplus \mask_\f] \prop} \infer[pvs-frame] {}{\propB * \pvs[\mask_1][\mask_2]\prop \proves \pvs[\mask_1][\mask_2] \propB * \prop} \inferH{pvs-allocI} {\text{$\mask$ is infinite}} {\later\prop \proves \pvs[\mask] \Exists \iname \in \mask. \knowInv\iname\prop} \inferH{pvs-openI} {}{\knowInv\iname\prop \proves \pvs[\set\iname][\emptyset] \later\prop} \inferH{pvs-closeI} {}{\knowInv\iname\prop \land \later\prop \proves \pvs[\emptyset][\set\iname] \TRUE} \inferH{pvs-update} {\melt \mupd \meltsB} {\ownGGhost\melt \proves \pvs[\mask] \Exists\meltB\in\meltsB. \ownGGhost\meltB} \end{mathpar} \paragraph{Laws of weakest preconditions.} \begin{mathpar} \infer[wp-value] {}{\prop[\val/\var] \proves \wpre{\val}[\mask]{\Ret\var.\prop}} \infer[wp-mono] {\mask_1 \subseteq \mask_2 \and \var:\textlog{val}\mid\prop \proves \propB} {\wpre\expr[\mask_1]{\Ret\var.\prop} \proves \wpre\expr[\mask_2]{\Ret\var.\propB}} \infer[pvs-wp] {}{\pvs[\mask] \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}} \infer[wp-pvs] {}{\wpre\expr[\mask]{\Ret\var.\pvs[\mask] \prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}} \infer[wp-atomic] {\mask_2 \subseteq \mask_1 \and \physatomic{\expr}} {\pvs[\mask_1][\mask_2] \wpre\expr[\mask_2]{\Ret\var. \pvs[\mask_2][\mask_1]\prop} \proves \wpre\expr[\mask_1]{\Ret\var.\prop}} \infer[wp-frame] {}{\propB * \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\propB*\prop}} \infer[wp-frame-step] {\toval(\expr) = \bot \and \mask_2 \subseteq \mask_1} {\wpre\expr[\mask]{\Ret\var.\prop} * \pvs[\mask_1][\mask_2]\later\pvs[\mask_2][\mask_1]\propB \proves \wpre\expr[\mask \uplus \mask_1]{\Ret\var.\propB*\prop}} \infer[wp-bind] {\text{$\lctx$ is a context}} {\wpre\expr[\mask]{\Ret\var. \wpre{\lctx(\ofval(\var))}[\mask]{\Ret\varB.\prop}} \proves \wpre{\lctx(\expr)}[\mask]{\Ret\varB.\prop}} \end{mathpar} \paragraph{Lifting of operational semantics.}~ \begin{mathpar} \infer[wp-lift-step] {\mask_2 \subseteq \mask_1 \and \toval(\expr_1) = \bot} { {\begin{inbox} % for some crazy reason, LaTeX is actually sensitive to the space between the "{ {" here and the "} }" below... ~~\pvs[\mask_1][\mask_2] \Exists \state_1. \red(\expr_1,\state_1) \land \later\ownPhys{\state_1} * {}\\\qquad\qquad\qquad \later\All \expr_2, \state_2, \expr_\f. \left( (\expr_1, \state_1 \step \expr_2, \state_2, \expr_\f) \land \ownPhys{\state_2} \right) \wand \pvs[\mask_2][\mask_1] \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_\f}[\top]{\Ret\any.\TRUE} {}\\\proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop} \end{inbox}} } \\\\ \infer[wp-lift-pure-step] {\toval(\expr_1) = \bot \and \All \state_1. \red(\expr_1, \state_1) \and \All \state_1, \expr_2, \state_2, \expr_\f. \expr_1,\state_1 \step \expr_2,\state_2,\expr_\f \Ra \state_1 = \state_2 } {\later\All \state, \expr_2, \expr_\f. (\expr_1,\state \step \expr_2, \state,\expr_\f) \Ra \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \wpre{\expr_\f}[\top]{\Ret\any.\TRUE} \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}} \end{mathpar} Notice that primitive view shifts cover everything to their right, \ie $\pvs \prop * \propB \eqdef \pvs (\prop * \propB)$. Here we define $\wpre{\expr_\f}[\mask]{\Ret\var.\prop} \eqdef \TRUE$ if $\expr_\f = \bot$ (remember that our stepping relation can, but does not have to, define a forked-off expression). The adequacy statement concerning functional correctness reads as follows: \begin{align*} &\All \mask, \expr, \val, \pred, \state, \melt, \state', \tpool'. \\&(\All n. \melt \in \mval_n) \Ra \\&( \ownPhys\state * \ownGGhost\melt \proves \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra \\&\cfg{\state}{[\expr]} \step^\ast \cfg{\state'}{[\val] \dplus \tpool'} \Ra \\&\pred(\val) \end{align*} where $\pred$ is a \emph{meta-level} predicate over values, \ie it can mention neither resources nor invariants. Furthermore, the following adequacy statement shows that our weakest preconditions imply that the execution never gets \emph{stuck}: Every expression in the thread pool either is a value, or can reduce further. \begin{align*} &\All \mask, \expr, \state, \melt, \state', \tpool'. \\&(\All n. \melt \in \mval_n) \Ra \\&( \ownPhys\state * \ownGGhost\melt \proves \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra \\&\cfg{\state}{[\expr]} \step^\ast \cfg{\state'}{\tpool'} \Ra \\&\All\expr'\in\tpool'. \toval(\expr') \neq \bot \lor \red(\expr', \state') \end{align*} Notice that this is stronger than saying that the thread pool can reduce; we actually assert that \emph{every} non-finished thread can take a step.  Ralf Jung committed Oct 04, 2016 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 \subsection{Iris model} \paragraph{Semantic domain of assertions.} \paragraph{Interpretation of assertions.} $\iProp$ is a $\UPred$, and hence the definitions from \Sref{sec:upred-logic} apply. We only have to define the interpretation of the missing connectives, the most interesting bits being primitive view shifts and weakest preconditions. \typedsection{World satisfaction}{\wsat{-}{-}{-} : \Delta\textdom{State} \times \Delta\pset{\mathbb{N}} \times \textdom{Res} \nfn \SProp } \begin{align*} \wsatpre(n, \mask, \state, \rss, \rs) & \eqdef \begin{inbox}[t] \rs \in \mval_{n+1} \land \rs.\pres = \exinj(\sigma) \land \dom(\rss) \subseteq \mask \cap \dom( \rs.\wld) \land {}\\ \All\iname \in \mask, \prop \in \iProp. (\rs.\wld)(\iname) \nequiv{n+1} \aginj(\latertinj(\wIso(\prop))) \Ra n \in \prop(\rss(\iname)) \end{inbox}\\ \wsat{\state}{\mask}{\rs} &\eqdef \set{0}\cup\setComp{n+1}{\Exists \rss : \mathbb{N} \fpfn \textdom{Res}. \wsatpre(n, \mask, \state, \rss, \rs \mtimes \prod_\iname \rss(\iname))} \end{align*} \typedsection{Primitive view-shift}{\mathit{pvs}_{-}^{-}(-) : \Delta(\pset{\mathbb{N}}) \times \Delta(\pset{\mathbb{N}}) \times \iProp \nfn \iProp} \begin{align*} \mathit{pvs}_{\mask_1}^{\mask_2}(\prop) &= \Lam \rs. \setComp{n}{\begin{aligned} \All \rs_\f, k, \mask_\f, \state.& 0 < k \leq n \land (\mask_1 \cup \mask_2) \disj \mask_\f \land k \in \wsat\state{\mask_1 \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\& \Exists \rsB. k \in \prop(\rsB) \land k \in \wsat\state{\mask_2 \cup \mask_\f}{\rsB \mtimes \rs_\f} \end{aligned}} \end{align*} \typedsection{Weakest precondition}{\mathit{wp}_{-}(-, -) : \Delta(\pset{\mathbb{N}}) \times \Delta(\textdom{Exp}) \times (\Delta(\textdom{Val}) \nfn \iProp) \nfn \iProp} $\textdom{wp}$ is defined as the fixed-point of a contractive function. \begin{align*} \textdom{pre-wp}(\textdom{wp})(\mask, \expr, \pred) &\eqdef \Lam\rs. \setComp{n}{\begin{aligned} \All &\rs_\f, m, \mask_\f, \state. 0 \leq m < n \land \mask \disj \mask_\f \land m+1 \in \wsat\state{\mask \cup \mask_\f}{\rs \mtimes \rs_\f} \Ra {}\\ &(\All\val. \toval(\expr) = \val \Ra \Exists \rsB. m+1 \in \pred(\val)(\rsB) \land m+1 \in \wsat\state{\mask \cup \mask_\f}{\rsB \mtimes \rs_\f}) \land {}\\ &(\toval(\expr) = \bot \land 0 < m \Ra \red(\expr, \state) \land \All \expr_2, \state_2, \expr_\f. \expr,\state \step \expr_2,\state_2,\expr_\f \Ra {}\\ &\qquad \Exists \rsB_1, \rsB_2. m \in \wsat\state{\mask \cup \mask_\f}{\rsB_1 \mtimes \rsB_2 \mtimes \rs_\f} \land m \in \textdom{wp}(\mask, \expr_2, \pred)(\rsB_1) \land {}&\\ &\qquad\qquad (\expr_\f = \bot \lor m \in \textdom{wp}(\top, \expr_\f, \Lam\any.\Lam\any.\mathbb{N})(\rsB_2)) \end{aligned}} \\ \textdom{wp}_\mask(\expr, \pred) &\eqdef \mathit{fix}(\textdom{pre-wp})(\mask, \expr, \pred) \end{align*} \typedsection{Interpretation of program logic assertions}{\Sem{\vctx \proves \term : \Prop} : \Sem{\vctx} \nfn \iProp} $\knowInv\iname\prop$, $\ownGGhost\melt$ and $\ownPhys\state$ are just syntactic sugar for forms of $\ownM{-}$. \begin{align*} \knowInv{\iname}{\prop} &\eqdef \ownM{[\iname \mapsto \aginj(\latertinj(\wIso(\prop)))], \munit, \munit} \\ \ownGGhost{\melt} &\eqdef \ownM{\munit, \munit, \melt} \\ \ownPhys{\state} &\eqdef \ownM{\munit, \exinj(\state), \munit} \\ ~\\ \Sem{\vctx \proves \pvs[\mask_1][\mask_2] \prop : \Prop}_\gamma &\eqdef \textdom{pvs}^{\Sem{\vctx \proves \mask_2 : \textlog{InvMask}}_\gamma}_{\Sem{\vctx \proves \mask_1 : \textlog{InvMask}}_\gamma}(\Sem{\vctx \proves \prop : \Prop}_\gamma) \\ \Sem{\vctx \proves \wpre{\expr}[\mask]{\Ret\var.\prop} : \Prop}_\gamma &\eqdef \textdom{wp}_{\Sem{\vctx \proves \mask : \textlog{InvMask}}_\gamma}(\Sem{\vctx \proves \expr : \textlog{Expr}}_\gamma, \Lam\val. \Sem{\vctx \proves \prop : \Prop}_{\gamma[\var\mapsto\val]}) \end{align*}  Ralf Jung committed Oct 04, 2016 305 306 307 308 309  %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: