program-logic.tex 30.2 KB
 Ralf Jung committed Oct 06, 2016 1 Ralf Jung committed Oct 04, 2016 2 \section{Program Logic} Ralf Jung committed Oct 06, 2016 3 \label{sec:program-logic} Ralf Jung committed Oct 04, 2016 4 Ralf Jung committed Oct 22, 2016 5 This section describes how to build a program logic for an arbitrary language (\cf \Sref{sec:language}) on top of the base logic. Ralf Jung committed Oct 06, 2016 6 So in the following, we assume that some language $\Lang$ was fixed. Ralf Jung committed Oct 06, 2016 7 Ralf Jung committed Nov 04, 2016 8 \subsection{Dynamic Composeable Higher-Order Resources} Ralf Jung committed Oct 22, 2016 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 \label{sec:composeable-resources} The base logic described in \Sref{sec:base-logic} works over an arbitrary CMRA $\monoid$ defining the structure of the resources. It turns out that we can generalize this further and permit picking CMRAs $\iFunc(\Prop)$'' that depend on the structure of assertions themselves. Of course, $\Prop$ is just the syntactic type of assertions; for this to make sense we have to look at the semantics. Furthermore, there is a composeability problem with the given logic: if we have one proof performed with CMRA $\monoid_1$, and another proof carried out with a \emph{different} CMRA $\monoid_2$, then the two proofs are actually carried out in two \emph{entirely separate logics} and hence cannot be combined. Finally, in many cases just having a single instance'' of a CMRA available for reasoning is not enough. For example, when reasoning about a dynamically allocated data structure, every time a new instance of that data structure is created, we will want a fresh resource governing the state of this particular instance. While it would be possible to handle this problem whenever it comes up, it turns out to be useful to provide a general solution. The purpose of this section is to describe how we solve these issues. \paragraph{Picking the resources.} The key ingredient that we will employ on top of the base logic is to give some more fixed structure to the resources. Ralf Jung committed Dec 05, 2016 25 To instantiate the program logic, the user picks a family of locally contractive bifunctors $(\iFunc_i : \OFEs \to \CMRAs)_{i \in \mathcal{I}}$. Ralf Jung committed Oct 22, 2016 26 27 28 29 (This is in contrast to the base logic, where the user picks a single, fixed CMRA that has a unit.) From this, we construct the bifunctor defining the overall resources as follows: \begin{align*} Ralf Jung committed Oct 31, 2016 30 \mathcal G \eqdef{}& \nat \\ Ralf Jung committed Dec 05, 2016 31 \textdom{ResF}(\ofe^\op, \ofe) \eqdef{}& \prod_{i \in \mathcal I} \mathcal G \fpfn \iFunc_i(\ofe^\op, \ofe) Ralf Jung committed Oct 22, 2016 32 33 \end{align*} We will motivate both the use of a product and the finite partial function below. Ralf Jung committed Dec 05, 2016 34 $\textdom{ResF}(\ofe^\op, \ofe)$ is a CMRA by lifting the individual CMRAs pointwise, and it has a unit (using the empty finite partial functions). Ralf Jung committed Oct 22, 2016 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 62 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 Furthermore, since the $\iFunc_i$ are locally contractive, so is $\textdom{ResF}$. Now we can write down the recursive domain equation: $\iPreProp \cong \UPred(\textdom{ResF}(\iPreProp, \iPreProp))$ $\iPreProp$ is a COFE defined as the fixed-point of a locally contractive bifunctor. This fixed-point exists and is unique\footnote{We have not proven uniqueness in Coq.} by America and Rutten's theorem~\cite{America-Rutten:JCSS89,birkedal:metric-space}. We do not need to consider how the object is constructed. We only need the isomorphism, given by \begin{align*} \Res &\eqdef \textdom{ResF}(\iPreProp, \iPreProp) \\ \iProp &\eqdef \UPred(\Res) \\ \wIso &: \iProp \nfn \iPreProp \\ \wIso^{-1} &: \iPreProp \nfn \iProp \end{align*} Notice that $\iProp$ is the semantic model of assertions for the base logic described in \Sref{sec:base-logic} with $\Res$: $\Sem{\Prop} \eqdef \iProp = \UPred(\Res)$ Effectively, we just defined a way to instantiate the base logic with $\Res$ as the CMRA of resources, while providing a way for $\Res$ to depend on $\iPreProp$, which is isomorphic to $\Sem\Prop$. We thus obtain all the rules of \Sref{sec:base-logic}, and furthermore, we can use the maps $\wIso$ and $\wIso^{-1}$ \emph{in the logic} to convert between logical assertions $\Sem\Prop$ and the domain $\iPreProp$ which is used in the construction of $\Res$ -- so from elements of $\iPreProp$, we can construct elements of $\Sem{\textlog M}$, which are the elements that can be owned in our logic. \paragraph{Proof composeability.} To make our proofs composeable, we \emph{generalize} our proofs over the family of functors. This is possible because we made $\Res$ a \emph{product} of all the CMRAs picked by the user, and because we can actually work with that product pointwise''. So instead of picking a \emph{concrete} family, proofs will assume to be given an \emph{arbitrary} family of functors, plus a proof that this family \emph{contains the functors they need}. Composing two proofs is then merely a matter of conjoining the assumptions they make about the functors. Since the logic is entirely parametric in the choice of functors, there is no trouble reasoning without full knowledge of the family of functors. Only when the top-level proof is completed we will close'' the proof by picking a concrete family that contains exactly those functors the proof needs. \paragraph{Dynamic resources.} Finally, the use of finite partial functions lets us have as many instances of any CMRA as we could wish for: Because there can only ever be finitely many instances already allocated, it is always possible to create a fresh instance with any desired (valid) starting state. This is best demonstrated by giving some proof rules. So let us first define the notion of ghost ownership that we use in this logic. Assuming that the family of functors contains the functor $\Sigma_i$ at index $i$, and furthermore assuming that $\monoid_i = \Sigma_i(\iPreProp, \iPreProp)$, given some $\melt \in \monoid_i$ we define: $\ownGhost\gname{\melt:\monoid_i} \eqdef \ownM{(\ldots, \emptyset, i:\mapsingleton \gname \melt, \emptyset, \ldots)}$ This is ownership of the pair (element of the product over all the functors) that has the empty finite partial function in all components \emph{except for} the component corresponding to index $i$, where we own the element $\melt$ at index $\gname$ in the finite partial function. We can show the following properties for this form of ownership: \begin{mathparpagebreakable} \inferH{res-alloc}{\text{$G$ infinite} \and \melt \in \mval_{M_i}} { \TRUE \proves \upd \Exists\gname\in G. \ownGhost\gname{\melt : M_i} } \and \inferH{res-update} {\melt \mupd_{M_i} B} {\ownGhost\gname{\melt : M_i} \proves \upd \Exists \meltB\in B. \ownGhost\gname{\meltB : M_i}} \inferH{res-empty} {\text{$\munit$ is a unit of $M_i$}} {\TRUE \proves \upd \ownGhost\gname\munit} \axiomH{res-op} {\ownGhost\gname{\melt : M_i} * \ownGhost\gname{\meltB : M_i} \provesIff \ownGhost\gname{\melt\mtimes\meltB : M_i}} \axiomH{res-valid} {\ownGhost\gname{\melt : M_i} \Ra \mval_{M_i}(\melt)} \inferH{res-timeless} Ralf Jung committed Dec 05, 2016 96 {\text{$\melt$ is a discrete OFE element}} Ralf Jung committed Oct 22, 2016 97 98 99 100 101 102 103 104 105 {\timeless{\ownGhost\gname{\melt : M_i}}} \end{mathparpagebreakable} Below, we will always work within (an instance of) the logic as described here. Whenever a CMRA is used in a proof, we implicitly assume it to be available in the global family of functors. We will typically leave the $M_i$ implicit when asserting ghost ownership, as the type of $\melt$ will be clear from the context. Ralf Jung committed Oct 21, 2016 106 \subsection{World Satisfaction, Invariants, Fancy Updates} Ralf Jung committed Oct 10, 2016 107 \label{sec:invariants} Ralf Jung committed Oct 06, 2016 108 109 110 111 112 113 114 To introduce invariants into our logic, we will define weakest precondition to explicitly thread through the proof that all the invariants are maintained throughout program execution. However, in order to be able to access invariants, we will also have to provide a way to \emph{temporarily disable} (or open'') them. To this end, we use tokens that manage which invariants are currently enabled. We assume to have the following four CMRAs available: \begin{align*} Ralf Jung committed Oct 31, 2016 115 116 117 118 119 \mathcal I \eqdef{}& \nat \\ \textmon{Inv} \eqdef{}& \authm(\mathcal I \fpfn \agm(\latert \iPreProp)) \\ \textmon{En} \eqdef{}& \pset{\mathcal I} \\ \textmon{Dis} \eqdef{}& \finpset{\mathcal I} \\ \textmon{State} \eqdef{}& \authm(\maybe{\exm(\State)}) Ralf Jung committed Oct 06, 2016 120 121 122 123 \end{align*} The last two are the tokens used for managing invariants, $\textmon{Inv}$ is the monoid used to manage the invariants themselves. Finally, $\textmon{State}$ is used to provide the program with a view of the physical state of the machine. Ralf Jung committed Dec 05, 2016 124 We assume that at the beginning of the verification, instances named $\gname_{\textmon{State}}$, $\gname_{\textmon{Inv}}$, $\gname_{\textmon{En}}$ and $\gname_{\textmon{Dis}}$ of these CMRAs have been created, such that these names are globally known. Ralf Jung committed Oct 06, 2016 125 Ralf Jung committed Oct 06, 2016 126 \paragraph{World Satisfaction.} Ralf Jung committed Oct 06, 2016 127 128 We can now define the assertion $W$ (\emph{world satisfaction}) which ensures that the enabled invariants are actually maintained: \begin{align*} Ralf Jung committed Oct 31, 2016 129 130 W \eqdef{}& \Exists I : \mathcal I \fpfn \Prop. \begin{array}[t]{@{} l} Robbert Krebbers committed Oct 17, 2016 131 \ownGhost{\gname_{\textmon{Inv}}}{\authfull Ralf Jung committed Oct 31, 2016 132 \mapComp {\iname} Robbert Krebbers committed Oct 17, 2016 133 134 135 136 {\aginj(\latertinj(\wIso(I(\iname))))} {\iname \in \dom(I)}} * \\ \Sep_{\iname \in \dom(I)} \left( \later I(\iname) * \ownGhost{\gname_{\textmon{Dis}}}{\set{\iname}} \lor \ownGhost{\gname_{\textmon{En}}}{\set{\iname}} \right) \end{array} Ralf Jung committed Oct 06, 2016 137 138 139 140 \end{align*} \paragraph{Invariants.} The following assertion states that an invariant with name $\iname$ exists and maintains assertion $\prop$: Robbert Krebbers committed Oct 17, 2016 141 142 $\knowInv\iname\prop \eqdef \ownGhost{\gname_{\textmon{Inv}}} {\authfrag \mapsingleton \iname {\aginj(\latertinj(\wIso(\prop)))}}$ Ralf Jung committed Oct 06, 2016 143 Ralf Jung committed Oct 21, 2016 144 145 \paragraph{Fancy Updates and View Shifts.} Next, we define \emph{fancy updates}, which are essentially the same as the basic updates of the base logic ($\Sref{sec:base-logic}$), except that they also have access to world satisfaction and can enable and disable invariants: Ralf Jung committed Oct 10, 2016 146 $\pvs[\mask_1][\mask_2] \prop \eqdef W * \ownGhost{\gname_{\textmon{En}}}{\mask_1} \wand \upd\diamond (W * \ownGhost{\gname_{\textmon{En}}}{\mask_2} * \prop)$ Ralf Jung committed Oct 06, 2016 147 Here, $\mask_1$ and $\mask_2$ are the \emph{masks} of the view update, defining which invariants have to be (at least!) available before and after the update. Robbert Krebbers committed Oct 17, 2016 148 We use $\top$ as symbol for the largest possible mask, $\nat$, and $\bot$ for the smallest possible mask $\emptyset$. Ralf Jung committed Oct 10, 2016 149 150 We will write $\pvs[\mask] \prop$ for $\pvs[\mask][\mask]\prop$. % Ralf Jung committed Oct 21, 2016 151 Fancy updates satisfy the following basic proof rules: Ralf Jung committed Oct 31, 2016 152 \begin{mathparpagebreakable} Ralf Jung committed Oct 21, 2016 153 \infer[fup-mono] Ralf Jung committed Oct 10, 2016 154 155 156 {\prop \proves \propB} {\pvs[\mask_1][\mask_2] \prop \proves \pvs[\mask_1][\mask_2] \propB} Ralf Jung committed Oct 21, 2016 157 \infer[fup-intro-mask] Ralf Jung committed Oct 10, 2016 158 {\mask_2 \subseteq \mask_1} Ralf Jung committed Oct 10, 2016 159 {\prop \proves \pvs[\mask_1][\mask_2]\pvs[\mask_2][\mask_1] \prop} Ralf Jung committed Oct 10, 2016 160 Ralf Jung committed Oct 21, 2016 161 \infer[fup-trans] Ralf Jung committed Oct 10, 2016 162 163 164 {} {\pvs[\mask_1][\mask_2] \pvs[\mask_2][\mask_3] \prop \proves \pvs[\mask_1][\mask_3] \prop} Ralf Jung committed Oct 21, 2016 165 \infer[fup-upd] Ralf Jung committed Oct 10, 2016 166 167 {}{\upd\prop \proves \pvs[\mask] \prop} Ralf Jung committed Oct 21, 2016 168 \infer[fup-frame] Ralf Jung committed Oct 10, 2016 169 {}{\propB * \pvs[\mask_1][\mask_2]\prop \proves \pvs[\mask_1 \uplus \mask_\f][\mask_2 \uplus \mask_\f] \propB * \prop} Ralf Jung committed Oct 10, 2016 170 Ralf Jung committed Oct 21, 2016 171 \inferH{fup-update} Ralf Jung committed Oct 10, 2016 172 173 174 {\melt \mupd \meltsB} {\ownM\melt \proves \pvs[\mask] \Exists\meltB\in\meltsB. \ownM\meltB} Ralf Jung committed Oct 21, 2016 175 \infer[fup-timeless] Ralf Jung committed Oct 10, 2016 176 177 {\timeless\prop} {\later\prop \proves \pvs[\mask] \prop} Ralf Jung committed Oct 10, 2016 178 % Ralf Jung committed Oct 21, 2016 179 % \inferH{fup-allocI} Ralf Jung committed Oct 10, 2016 180 181 182 % {\text{$\mask$ is infinite}} % {\later\prop \proves \pvs[\mask] \Exists \iname \in \mask. \knowInv\iname\prop} %gov Ralf Jung committed Oct 21, 2016 183 % \inferH{fup-openI} Ralf Jung committed Oct 10, 2016 184 185 % {}{\knowInv\iname\prop \proves \pvs[\set\iname][\emptyset] \later\prop} % Ralf Jung committed Oct 21, 2016 186 % \inferH{fup-closeI} Ralf Jung committed Oct 10, 2016 187 % {}{\knowInv\iname\prop \land \later\prop \proves \pvs[\emptyset][\set\iname] \TRUE} Ralf Jung committed Oct 31, 2016 188 \end{mathparpagebreakable} Ralf Jung committed Dec 05, 2016 189 (There are no rules related to invariants here. Those rules will be discussed later, in \Sref{sec:namespaces}.) Ralf Jung committed Oct 06, 2016 190 Ralf Jung committed Oct 21, 2016 191 We can further define the notions of \emph{view shifts} and \emph{linear view shifts}: Ralf Jung committed Oct 06, 2016 192 \begin{align*} Ralf Jung committed Oct 21, 2016 193 194 \prop \vsW[\mask_1][\mask_2] \propB \eqdef{}& \prop \wand \pvs[\mask_1][\mask_2] \propB \\ \prop \vs[\mask_1][\mask_2] \propB \eqdef{}& \always(\prop \wand \pvs[\mask_1][\mask_2] \propB) Ralf Jung committed Oct 06, 2016 195 \end{align*} Ralf Jung committed Oct 21, 2016 196 These two are useful when writing down specifications and for comparing with previous versions of Iris, but for reasoning, it is typically easier to just work directly with fancy updates. Ralf Jung committed Oct 10, 2016 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 Still, just to give an idea of what view shifts are'', here are some proof rules for them: \begin{mathparpagebreakable} \inferH{vs-update} {\melt \mupd \meltsB} {\ownGhost\gname{\melt} \vs \exists \meltB \in \meltsB.\; \ownGhost\gname{\meltB}} \and \inferH{vs-trans} {\prop \vs[\mask_1][\mask_2] \propB \and \propB \vs[\mask_2][\mask_3] \propC} {\prop \vs[\mask_1][\mask_3] \propC} \and \inferH{vs-imp} {\always{(\prop \Ra \propB)}} {\prop \vs[\emptyset] \propB} \and \inferH{vs-mask-frame} {\prop \vs[\mask_1][\mask_2] \propB} {\prop \vs[\mask_1 \uplus \mask'][\mask_2 \uplus \mask'] \propB} \and \inferH{vs-frame} {\prop \vs[\mask_1][\mask_2] \propB} {\prop * \propC \vs[\mask_1][\mask_2] \propB * \propC} \and \inferH{vs-timeless} {\timeless{\prop}} {\later \prop \vs \prop} Ralf Jung committed Oct 10, 2016 223 224 225 226 227 228 229 230 231 232 % \inferH{vs-allocI} % {\infinite(\mask)} % {\later{\prop} \vs[\mask] \exists \iname\in\mask.\; \knowInv{\iname}{\prop}} % \and % \axiomH{vs-openI} % {\knowInv{\iname}{\prop} \proves \TRUE \vs[\{ \iname \} ][\emptyset] \later \prop} % \and % \axiomH{vs-closeI} % {\knowInv{\iname}{\prop} \proves \later \prop \vs[\emptyset][\{ \iname \} ] \TRUE } % Ralf Jung committed Oct 10, 2016 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 \inferHB{vs-disj} {\prop \vs[\mask_1][\mask_2] \propC \and \propB \vs[\mask_1][\mask_2] \propC} {\prop \lor \propB \vs[\mask_1][\mask_2] \propC} \and \inferHB{vs-exist} {\All \var. (\prop \vs[\mask_1][\mask_2] \propB)} {(\Exists \var. \prop) \vs[\mask_1][\mask_2] \propB} \and \inferHB{vs-always} {\always\propB \proves \prop \vs[\mask_1][\mask_2] \propC} {\prop \land \always{\propB} \vs[\mask_1][\mask_2] \propC} \and \inferH{vs-false} {} {\FALSE \vs[\mask_1][\mask_2] \prop } \end{mathparpagebreakable} \subsection{Weakest Precondition} Ralf Jung committed Oct 06, 2016 251 252 Finally, we can define the core piece of the program logic, the assertion that reasons about program behavior: Weakest precondition, from which Hoare triples will be derived. Ralf Jung committed Oct 10, 2016 253 254 \paragraph{Defining weakest precondition.} Ralf Jung committed Oct 06, 2016 255 256 257 We assume that everything making up the definition of the language, \ie values, expressions, states, the conversion functions, reduction relation and all their properties, are suitably reflected into the logic (\ie they are part of the signature $\Sig$). \begin{align*} Ralf Jung committed Oct 06, 2016 258 259 260 \textdom{wp} \eqdef{}& \MU \textdom{wp}. \Lam \mask, \expr, \pred. \\ & (\Exists\val. \toval(\expr) = \val \land \pvs[\mask] \prop) \lor {}\\ & \Bigl(\toval(\expr) = \bot \land \All \state. \ownGhost{\gname_{\textmon{State}}}{\authfull \state} \vsW[\mask][\emptyset] {}\\ Robbert Krebbers committed Oct 17, 2016 261 262 &\qquad \red(\expr, \state) * \later\All \expr', \state', \vec\expr. (\expr, \state \step \expr', \state', \vec\expr) \vsW[\emptyset][\mask] {}\\ &\qquad\qquad \ownGhost{\gname_{\textmon{State}}}{\authfull \state'} * \textdom{wp}(\mask, \expr', \pred) * \Sep_{\expr'' \in \vec\expr} \textdom{wp}(\top, \expr'', \Lam \any. \TRUE)\Bigr) \\ Ralf Jung committed Oct 06, 2016 263 % (* value case *) Ralf Jung committed Oct 06, 2016 264 \wpre\expr[\mask]{\Ret\val. \prop} \eqdef{}& \textdom{wp}(\mask, \expr, \Lam\val.\prop) Ralf Jung committed Oct 06, 2016 265 \end{align*} Ralf Jung committed Oct 06, 2016 266 If we leave away the mask, we assume it to default to $\top$. Ralf Jung committed Oct 06, 2016 267 Ralf Jung committed Oct 06, 2016 268 269 270 271 This ties the authoritative part of \textmon{State} to the actual physical state of the reduction witnessed by the weakest precondition. The fragment will then be available to the user of the logic, as their way of talking about the physical state: $\ownPhys\state \eqdef \ownGhost{\gname_{\textmon{State}}}{\authfrag \state}$ Ralf Jung committed Oct 10, 2016 272 \paragraph{Laws of weakest precondition.} Ralf Jung committed Oct 22, 2016 273 The following rules can all be derived: Ralf Jung committed Oct 04, 2016 274 275 276 277 278 \begin{mathpar} \infer[wp-value] {}{\prop[\val/\var] \proves \wpre{\val}[\mask]{\Ret\var.\prop}} \infer[wp-mono] Ralf Jung committed Oct 10, 2016 279 280 {\mask_1 \subseteq \mask_2 \and \vctx,\var:\textlog{val}\mid\prop \proves \propB} {\vctx\mid\wpre\expr[\mask_1]{\Ret\var.\prop} \proves \wpre\expr[\mask_2]{\Ret\var.\propB}} Ralf Jung committed Oct 04, 2016 281 Ralf Jung committed Oct 21, 2016 282 \infer[fup-wp] Ralf Jung committed Oct 04, 2016 283 284 {}{\pvs[\mask] \wpre\expr[\mask]{\Ret\var.\prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}} Ralf Jung committed Oct 21, 2016 285 \infer[wp-fup] Ralf Jung committed Oct 04, 2016 286 287 288 {}{\wpre\expr[\mask]{\Ret\var.\pvs[\mask] \prop} \proves \wpre\expr[\mask]{\Ret\var.\prop}} \infer[wp-atomic] Ralf Jung committed Oct 10, 2016 289 {\physatomic{\expr}} Ralf Jung committed Oct 04, 2016 290 291 292 293 294 295 296 297 {\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} Ralf Jung committed Oct 10, 2016 298 {\wpre\expr[\mask_2]{\Ret\var.\prop} * \pvs[\mask_1][\mask_2]\later\pvs[\mask_2][\mask_1]\propB \proves \wpre\expr[\mask_1]{\Ret\var.\propB*\prop}} Ralf Jung committed Oct 04, 2016 299 300 301 302 303 304 \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} Ralf Jung committed Oct 10, 2016 305 306 We will also want rules that connect weakest preconditions to the operational semantics of the language. In order to cover the most general case, those rules end up being more complicated: Ralf Jung committed Oct 04, 2016 307 308 \begin{mathpar} \infer[wp-lift-step] Ralf Jung committed Oct 21, 2016 309 {} Ralf Jung committed Oct 04, 2016 310 { {\begin{inbox} % for some crazy reason, LaTeX is actually sensitive to the space between the "{ {" here and the "} }" below... Robbert Krebbers committed Oct 17, 2016 311 ~~\pvs[\mask][\emptyset] \Exists \state_1. \red(\expr_1,\state_1) * \later\ownPhys{\state_1} * {}\\\qquad~~ \later\All \expr_2, \state_2, \vec\expr. \Bigl( (\expr_1, \state_1 \step \expr_2, \state_2, \vec\expr) * \ownPhys{\state_2} \Bigr) \wand \pvs[\emptyset][\mask] \Bigl(\wpre{\expr_2}[\mask]{\Ret\var.\prop} * \Sep_{\expr_\f \in \vec\expr} \wpre{\expr_\f}[\top]{\Ret\any.\TRUE}\Bigr) {}\\\proves \wpre{\expr_1}[\mask]{\Ret\var.\prop} Ralf Jung committed Oct 04, 2016 312 313 314 \end{inbox}} } \\\\ \infer[wp-lift-pure-step] Ralf Jung committed Oct 21, 2016 315 {\All \state_1. \red(\expr_1, \state_1) \and Robbert Krebbers committed Oct 17, 2016 316 317 \All \state_1, \expr_2, \state_2, \vec\expr. \expr_1,\state_1 \step \expr_2,\state_2,\vec\expr \Ra \state_1 = \state_2 } {\later\All \state, \expr_2, \vec\expr. (\expr_1,\state \step \expr_2, \state,\vec\expr) \Ra \wpre{\expr_2}[\mask]{\Ret\var.\prop} * \Sep_{\expr_\f \in \vec\expr} \wpre{\expr_\f}[\top]{\Ret\any.\TRUE} \proves \wpre{\expr_1}[\mask]{\Ret\var.\prop}} Ralf Jung committed Oct 04, 2016 318 319 \end{mathpar} Ralf Jung committed Oct 10, 2016 320 321 322 323 324 325 We can further derive some slightly simpler rules for special cases: We can derive some specialized forms of the lifting axioms for the operational semantics. \begin{mathparpagebreakable} \infer[wp-lift-atomic-step] {\atomic(\expr_1) \and \red(\expr_1, \state_1)} Robbert Krebbers committed Oct 17, 2016 326 { {\begin{inbox}~~\later\ownPhys{\state_1} * \later\All \val_2, \state_2, \vec\expr. (\expr_1,\state_1 \step \ofval(\val),\state_2,\vec\expr) * \ownPhys{\state_2} \wand \prop[\val_2/\var] * \Sep_{\expr_\f \in \vec\expr} \wpre{\expr_\f}[\top]{\Ret\any.\TRUE} {}\\ \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop} Ralf Jung committed Oct 10, 2016 327 328 329 330 331 \end{inbox}} } \infer[wp-lift-atomic-det-step] {\atomic(\expr_1) \and \red(\expr_1, \state_1) \and Robbert Krebbers committed Oct 17, 2016 332 333 \All \expr'_2, \state'_2, \vec\expr'. \expr_1,\state_1 \step \expr'_2,\state'_2,\vec\expr' \Ra \state_2 = \state_2' \land \toval(\expr_2') = \val_2 \land \vec\expr = \vec\expr'} {\later\ownPhys{\state_1} * \later \Bigl(\ownPhys{\state_2} \wand \prop[\val_2/\var] * \Sep_{\expr_\f \in \vec\expr} \wpre{\expr_\f}[\top]{\Ret\any.\TRUE} \Bigr) \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}} Ralf Jung committed Oct 10, 2016 334 335 \infer[wp-lift-pure-det-step] Ralf Jung committed Oct 21, 2016 336 {\All \state_1. \red(\expr_1, \state_1) \\ Robbert Krebbers committed Oct 17, 2016 337 338 \All \state_1, \expr_2', \state'_2, \vec\expr'. \expr_1,\state_1 \step \expr'_2,\state'_2,\vec\expr' \Ra \state_1 = \state'_2 \land \expr_2 = \expr_2' \land \vec\expr = \vec\expr'} {\later \Bigl( \wpre{\expr_2}[\mask_1]{\Ret\var.\prop} * \Sep_{\expr_\f \in \vec\expr} \wpre{\expr_\f}[\top]{\Ret\any.\TRUE} \Bigr) \proves \wpre{\expr_1}[\mask_1]{\Ret\var.\prop}} Ralf Jung committed Oct 10, 2016 339 \end{mathparpagebreakable} Ralf Jung committed Oct 10, 2016 340 341 Ralf Jung committed Oct 10, 2016 342 \paragraph{Adequacy of weakest precondition.} Ralf Jung committed Oct 04, 2016 343 Ralf Jung committed Oct 10, 2016 344 345 346 347 The purpose of the adequacy statement is to show that our notion of weakest preconditions is \emph{realistic} in the sense that it actually has anything to do with the actual behavior of the program. There are two properties we are looking for: First of all, the postcondition should reflect actual properties of the values the program can terminate with. Second, a proof of a weakest precondition with any postcondition should imply that the program is \emph{safe}, \ie that it does not get stuck. Robbert Krebbers committed Oct 17, 2016 348 To express the adequacy statement for functional correctness, we assume we are given some set $V \subseteq \Val$ of legal return values. Ralf Jung committed Oct 10, 2016 349 350 Furthermore, we assume that the signature $\Sig$ adds a predicate $\pred$ to the logic which reflects $V$ into the logic: $\begin{array}{rMcMl} Robbert Krebbers committed Oct 17, 2016 351 \Sem\pred &:& \Sem{\Val\,} \nfn \Sem\Prop \\ Ralf Jung committed Oct 10, 2016 352 353 354 355 356 \Sem\pred &\eqdef& \Lam \val. \Lam \any. \setComp{n}{v \in V} \end{array}$ The signature can of course state arbitrary additional properties of $\pred$, as long as they are proven sound. The adequacy statement now reads as follows: Ralf Jung committed Oct 04, 2016 357 \begin{align*} Ralf Jung committed Oct 10, 2016 358 359 &\All \mask, \expr, \val, \pred, \state, \state', \tpool'. \\&( \ownPhys\state \proves \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra Ralf Jung committed Oct 04, 2016 360 361 \\&\cfg{\state}{[\expr]} \step^\ast \cfg{\state'}{[\val] \dplus \tpool'} \Ra Ralf Jung committed Oct 10, 2016 362 \\&\val \in V Ralf Jung committed Oct 04, 2016 363 364 \end{align*} Ralf Jung committed Oct 10, 2016 365 The adequacy statement for safety says that our weakest preconditions imply that every expression in the thread pool either is a value, or can reduce further. Ralf Jung committed Oct 04, 2016 366 \begin{align*} Ralf Jung committed Oct 10, 2016 367 &\All \mask, \expr, \state, \state', \tpool'. Ralf Jung committed Oct 04, 2016 368 \\&(\All n. \melt \in \mval_n) \Ra Ralf Jung committed Oct 10, 2016 369 \\&( \ownPhys\state \proves \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra Ralf Jung committed Oct 04, 2016 370 371 372 373 374 375 \\&\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 10, 2016 376 377 378 379 \paragraph{Hoare triples.} It turns out that weakest precondition is actually quite convenient to work with, in particular when perfoming these proofs in Coq. Still, for a more traditional presentation, we can easily derive the notion of a Hoare triple: $Ralf Jung committed Oct 21, 2016 380 \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask] \eqdef \always{(\prop \wand \wpre{\expr}[\mask]{\Ret\val.\propB})} Ralf Jung committed Oct 10, 2016 381$ Ralf Jung committed Oct 04, 2016 382 Ralf Jung committed Oct 10, 2016 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 We only give some of the proof rules for Hoare triples here, since we usually do all our reasoning directly with weakest preconditions and use Hoare triples only to write specifications. \begin{mathparpagebreakable} \inferH{Ht-ret} {} {\hoare{\TRUE}{\valB}{\Ret\val. \val = \valB}[\mask]} \and \inferH{Ht-bind} {\text{$\lctx$ is a context} \and \hoare{\prop}{\expr}{\Ret\val. \propB}[\mask] \\ \All \val. \hoare{\propB}{\lctx(\val)}{\Ret\valB.\propC}[\mask]} {\hoare{\prop}{\lctx(\expr)}{\Ret\valB.\propC}[\mask]} \and \inferH{Ht-csq} {\prop \vs \prop' \\ \hoare{\prop'}{\expr}{\Ret\val.\propB'}[\mask] \\ \All \val. \propB' \vs \propB} {\hoare{\prop}{\expr}{\Ret\val.\propB}[\mask]} \and % \inferH{Ht-mask-weaken} % {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask]} % {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask \uplus \mask']} % \\\\ \inferH{Ht-frame} {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask]} {\hoare{\prop * \propC}{\expr}{\Ret\val. \propB * \propC}[\mask]} \and % \inferH{Ht-frame-step} % {\hoare{\prop}{\expr}{\Ret\val. \propB}[\mask] \and \toval(\expr) = \bot \and \mask_2 \subseteq \mask_2 \\\\ \propC_1 \vs[\mask_1][\mask_2] \later\propC_2 \and \propC_2 \vs[\mask_2][\mask_1] \propC_3} % {\hoare{\prop * \propC_1}{\expr}{\Ret\val. \propB * \propC_3}[\mask \uplus \mask_1]} % \and \inferH{Ht-atomic} {\prop \vs[\mask \uplus \mask'][\mask] \prop' \\ \hoare{\prop'}{\expr}{\Ret\val.\propB'}[\mask] \\ \All\val. \propB' \vs[\mask][\mask \uplus \mask'] \propB \\ \physatomic{\expr} } {\hoare{\prop}{\expr}{\Ret\val.\propB}[\mask \uplus \mask']} \and \inferH{Ht-false} {} {\hoare{\FALSE}{\expr}{\Ret \val. \prop}[\mask]} \and \inferHB{Ht-disj} {\hoare{\prop}{\expr}{\Ret\val.\propC}[\mask] \and \hoare{\propB}{\expr}{\Ret\val.\propC}[\mask]} {\hoare{\prop \lor \propB}{\expr}{\Ret\val.\propC}[\mask]} \and \inferHB{Ht-exist} {\All \var. \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask]} {\hoare{\Exists \var. \prop}{\expr}{\Ret\val.\propB}[\mask]} \and \inferHB{Ht-box} {\always\propB \proves \hoare{\prop}{\expr}{\Ret\val.\propC}[\mask]} {\hoare{\prop \land \always{\propB}}{\expr}{\Ret\val.\propC}[\mask]} % \and % \inferH{Ht-inv} % {\hoare{\later\propC*\prop}{\expr}{\Ret\val.\later\propC*\propB}[\mask] \and % \physatomic{\expr} % } % {\knowInv\iname\propC \proves \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask \uplus \set\iname]} % \and % \inferH{Ht-inv-timeless} % {\hoare{\propC*\prop}{\expr}{\Ret\val.\propC*\propB}[\mask] \and % \physatomic{\expr} \and \timeless\propC % } % {\knowInv\iname\propC \proves \hoare{\prop}{\expr}{\Ret\val.\propB}[\mask \uplus \set\iname]} \end{mathparpagebreakable} Ralf Jung committed Oct 04, 2016 448 Ralf Jung committed Oct 10, 2016 449 450 451 452 453 454 455 456 457 458 \subsection{Invariant Namespaces} \label{sec:namespaces} In \Sref{sec:invariants}, we defined an assertion $\knowInv\iname\prop$ expressing knowledge (\ie the assertion is persistent) that $\prop$ is maintained as invariant with name $\iname$. The concrete name $\iname$ is picked when the invariant is allocated, so it cannot possibly be statically known -- it will always be a variable that's threaded through everything. However, we hardly care about the actual, concrete name. All we need to know is that this name is \emph{different} from the names of other invariants that we want to open at the same time. Keeping track of the $n^2$ mutual inequalities that arise with $n$ invariants quickly gets in the way of the actual proof. To solve this issue, instead of remembering the exact name picked for an invariant, we will keep track of the \emph{namespace} the invariant was allocated in. Jacques-Henri Jourdan committed Oct 13, 2016 459 Namespaces are sets of invariants, following a tree-like structure: Ralf Jung committed Oct 10, 2016 460 461 462 463 464 465 466 467 Think of the name of an invariant as a sequence of identifiers, much like a fully qualified Java class name. A \emph{namespace} $\namesp$ then is like a Java package: it is a sequence of identifiers that we think of as \emph{containing} all invariant names that begin with this sequence. For example, \texttt{org.mpi-sws.iris} is a namespace containing the invariant name \texttt{org.mpi-sws.iris.heap}. The crux is that all namespaces contain infinitely many invariants, and hence we can \emph{freely pick} the namespace an invariant is allocated in -- no further, unpredictable choice has to be made. Furthermore, we will often know that namespaces are \emph{disjoint} just by looking at them. The namespaces $\namesp.\texttt{iris}$ and $\namesp.\texttt{gps}$ are disjoint no matter the choice of $\namesp$. As a result, there is often no need to track disjointness of namespaces, we just have to pick the namespaces that we allocate our invariants in accordingly. Robbert Krebbers committed Oct 17, 2016 468 Formally speaking, let $\namesp \in \textlog{InvNamesp} \eqdef \List(\nat)$ be the type of \emph{invariant namespaces}. Ralf Jung committed Oct 10, 2016 469 470 471 472 We use the notation $\namesp.\iname$ for the namespace $[\iname] \dplus \namesp$. (In other words, the list is backwards''. This is because cons-ing to the list, like the dot does above, is easier to deal with in Coq than appending at the end.) The elements of a namespaces are \emph{structured invariant names} (think: Java fully qualified class name). Robbert Krebbers committed Oct 17, 2016 473 They, too, are lists of $\nat$, the same type as namespaces. Ralf Jung committed Oct 31, 2016 474 475 In order to connect this up to the definitions of \Sref{sec:invariants}, we need a way to map structued invariant names to $\mathcal I$, the type of plain'' invariant names. Any injective mapping $\textlog{namesp\_inj}$ will do; and such a mapping has to exist because $\List(\nat)$ is countable and $\mathcal I$ is infinite. Ralf Jung committed Oct 10, 2016 476 477 478 479 Whenever needed, we (usually implicitly) coerce $\namesp$ to its encoded suffix-closure, \ie to the set of encoded structured invariant names contained in the namespace: $\namecl\namesp \eqdef \setComp{\iname}{\Exists \namesp'. \iname = \textlog{namesp\_inj}(\namesp' \dplus \namesp)}$ We will overload the notation for invariant assertions for using namespaces instead of names: $\knowInv\namesp\prop \eqdef \Exists \iname \in \namecl\namesp. \knowInv\iname{\prop}$ Ralf Jung committed Oct 21, 2016 480 We can now derive the following rules (this involves unfolding the definition of fancy updates): Ralf Jung committed Oct 10, 2016 481 482 \begin{mathpar} \axiomH{inv-persist}{\knowInv\namesp\prop \proves \always\knowInv\namesp\prop} Ralf Jung committed Oct 04, 2016 483 Jacques-Henri Jourdan committed Oct 13, 2016 484 \axiomH{inv-alloc}{\later\prop \proves \pvs[\emptyset] \knowInv\namesp\prop} Ralf Jung committed Oct 04, 2016 485 Ralf Jung committed Oct 10, 2016 486 487 488 \inferH{inv-open} {\namesp \subseteq \mask} {\knowInv\namesp\prop \vs[\mask][\mask\setminus\namesp] \later\prop * (\later\prop \vsW[\mask\setminus\namesp][\mask] \TRUE)} Ralf Jung committed Oct 04, 2016 489 Ralf Jung committed Oct 10, 2016 490 491 492 493 \inferH{inv-open-timeless} {\namesp \subseteq \mask \and \timeless\prop} {\knowInv\namesp\prop \vs[\mask][\mask\setminus\namesp] \prop * (\prop \vsW[\mask\setminus\namesp][\mask] \TRUE)} \end{mathpar} Ralf Jung committed Oct 04, 2016 494 Ralf Jung committed Oct 10, 2016 495 496 497 498 499 500 501 502 503 504 505 506 507 508 \subsection{Accessors} The two rules \ruleref{inv-open} and \ruleref{inv-open-timeless} above may look a little surprising, in the sense that it is not clear on first sight how they would be applied. The rules are the first \emph{accessors} that show up in this document. Accessors are assertions of the form $\prop \vs[\mask_1][\mask_2] \Exists\var. \propB * (\All\varB. \propB' \vsW[\mask_2][\mask_1] \propC)$ One way to think about such assertions is as follows: Given some accessor, if during our verification we have the assertion $\prop$ and the mask $\mask_1$ available, we can use the accessor to \emph{access} $\propB$ and obtain the witness $\var$. We call this \emph{opening} the accessor, and it changes the mask to $\mask_2$. Additionally, opening the accessor provides us with $\All\varB. \propB' \vsW[\mask_2][\mask_1] \propC$, a \emph{linear view shift} (\ie a view shift that can only be used once). This linear view shift tells us that in order to \emph{close} the accessor again and go back to mask $\mask_1$, we have to pick some $\varB$ and establish the corresponding $\propB'$. After closing, we will obtain $\propC$. Ralf Jung committed Oct 21, 2016 509 Using \ruleref{vs-trans} and \ruleref{Ht-atomic} (or the corresponding proof rules for fancy updates and weakest preconditions), we can show that it is possible to open an accessor around any view shift and any \emph{atomic} expression. Ralf Jung committed Oct 10, 2016 510 511 512 513 514 Furthermore, in the special case that $\mask_1 = \mask_2$, the accessor can be opened around \emph{any} expression. For this reason, we also call such accessors \emph{non-atomic}. The reasons accessors are useful is that they let us talk about opening X'' (\eg opening invariants'') without having to care what X is opened around. Furthermore, as we construct more sophisticated and more interesting things that can be opened (\eg invariants that can be cancelled'', or STSs), accessors become a useful interface that allows us to mix and match different abstractions in arbitrary ways. Ralf Jung committed Oct 04, 2016 515 Ralf Jung committed Dec 06, 2016 516 517 518 519 For the special case that $\prop = \propC$ and $\propB = \propB'$, we use the following notation that avoids repetition: $\Acc[\mask_1][\mask_2]\prop{\Ret x. \propB} \eqdef \prop \vs[\mask_1][\mask_2] \Exists\var. \propB * (\propB \vsW[\mask_2][\mask_1] \prop)$ This accessor is idempotent'' in the sense that it doesn't actually change the state. After applying it, we get our $\prop$ back so we end up where we started. Ralf Jung committed Oct 04, 2016 520 521 522 523 %%% Local Variables: %%% mode: latex %%% TeX-master: "iris" %%% End: