program-logic.tex 25.5 KB
Newer Older
Ralf Jung's avatar
Ralf Jung committed
1

2
\section{Program Logic}
3
\label{sec:program-logic}
4

Ralf Jung's avatar
Ralf Jung committed
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's avatar
Ralf Jung committed
6
So in the following, we assume that some language $\Lang$ was fixed.
7
Furthermore, we work in the logic with higher-order ghost state as described in \Sref{sec:composeable-resources}.
Ralf Jung's avatar
Ralf Jung committed
8 9


10
\subsection{World Satisfaction, Invariants, Fancy Updates}
Ralf Jung's avatar
Ralf Jung committed
11
\label{sec:invariants}
Ralf Jung's avatar
Ralf Jung committed
12 13 14 15 16

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.

17
We assume to have the following four cameras available:
Ralf Jung's avatar
Ralf Jung committed
18
\begin{align*}
19 20 21
  \InvName \eqdef{}& \nat \\
  \textmon{Inv} \eqdef{}& \authm(\InvName \fpfn \agm(\latert \iPreProp)) \\
  \textmon{En} \eqdef{}& \pset{\InvName} \\
22
  \textmon{Dis} \eqdef{}& \finpset{\InvName}
Ralf Jung's avatar
Ralf Jung committed
23 24 25
\end{align*}
The last two are the tokens used for managing invariants, $\textmon{Inv}$ is the monoid used to manage the invariants themselves.

26
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 cameras have been created, such that these names are globally known.
Ralf Jung's avatar
Ralf Jung committed
27

Ralf Jung's avatar
Ralf Jung committed
28
\paragraph{World Satisfaction.}
29
We can now define the proposition $W$ (\emph{world satisfaction}) which ensures that the enabled invariants are actually maintained:
Ralf Jung's avatar
Ralf Jung committed
30
\begin{align*}
31
  W \eqdef{}& \Exists I : \InvName \fpfn \Prop.
32
  \begin{array}[t]{@{} l}
33
    \ownGhost{\gname_{\textmon{Inv}}}{\authfull
Ralf Jung's avatar
Ralf Jung committed
34
      \mapComp {\iname}
35 36 37 38
        {\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's avatar
Ralf Jung committed
39 40 41
\end{align*}

\paragraph{Invariants.}
42
The following proposition states that an invariant with name $\iname$ exists and maintains proposition $\prop$:
43 44
\[ \knowInv\iname\prop \eqdef \ownGhost{\gname_{\textmon{Inv}}}
  {\authfrag \mapsingleton \iname {\aginj(\latertinj(\wIso(\prop)))}} \]
Ralf Jung's avatar
Ralf Jung committed
45

46
\paragraph{Fancy Updates and View Shifts.}
Ralf Jung's avatar
Ralf Jung committed
47
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:
48
\[ \pvs[\mask_1][\mask_2] \prop \eqdef W * \ownGhost{\gname_{\textmon{En}}}{\mask_1} \wand \upd\diamond (W * \ownGhost{\gname_{\textmon{En}}}{\mask_2} * \prop) \]
Ralf Jung's avatar
Ralf Jung committed
49
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's avatar
Robbert Krebbers committed
50
We use $\top$ as symbol for the largest possible mask, $\nat$, and $\bot$ for the smallest possible mask $\emptyset$.
51 52
We will write $\pvs[\mask] \prop$ for $\pvs[\mask][\mask]\prop$.
%
Ralf Jung's avatar
Ralf Jung committed
53
Fancy updates satisfy the following basic proof rules:
54
\begin{mathparpagebreakable}
Ralf Jung's avatar
Ralf Jung committed
55
\infer[fup-mono]
56 57 58
{\prop \proves \propB}
{\pvs[\mask_1][\mask_2] \prop \proves \pvs[\mask_1][\mask_2] \propB}

Ralf Jung's avatar
Ralf Jung committed
59
\infer[fup-intro-mask]
60
{\mask_2 \subseteq \mask_1}
61
{\prop \proves \pvs[\mask_1][\mask_2]\pvs[\mask_2][\mask_1] \prop}
62

Ralf Jung's avatar
Ralf Jung committed
63
\infer[fup-trans]
64 65 66
{}
{\pvs[\mask_1][\mask_2] \pvs[\mask_2][\mask_3] \prop \proves \pvs[\mask_1][\mask_3] \prop}

Ralf Jung's avatar
Ralf Jung committed
67
\infer[fup-upd]
68 69
{}{\upd\prop \proves \pvs[\mask] \prop}

Ralf Jung's avatar
Ralf Jung committed
70
\infer[fup-frame]
71
{}{\propB * \pvs[\mask_1][\mask_2]\prop \proves \pvs[\mask_1 \uplus \mask_\f][\mask_2 \uplus \mask_\f] \propB * \prop}
72

Ralf Jung's avatar
Ralf Jung committed
73
\inferH{fup-update}
74 75 76
{\melt \mupd \meltsB}
{\ownM\melt \proves \pvs[\mask] \Exists\meltB\in\meltsB. \ownM\meltB}

Ralf Jung's avatar
Ralf Jung committed
77
\infer[fup-timeless]
78 79
{\timeless\prop}
{\later\prop \proves \pvs[\mask] \prop}
80
%
Ralf Jung's avatar
Ralf Jung committed
81
% \inferH{fup-allocI}
82 83 84
% {\text{$\mask$ is infinite}}
% {\later\prop \proves \pvs[\mask] \Exists \iname \in \mask. \knowInv\iname\prop}
%gov
Ralf Jung's avatar
Ralf Jung committed
85
% \inferH{fup-openI}
86 87
% {}{\knowInv\iname\prop \proves \pvs[\set\iname][\emptyset] \later\prop}
%
Ralf Jung's avatar
Ralf Jung committed
88
% \inferH{fup-closeI}
89
% {}{\knowInv\iname\prop \land \later\prop \proves \pvs[\emptyset][\set\iname] \TRUE}
90
\end{mathparpagebreakable}
91
(There are no rules related to invariants here. Those rules will be discussed later, in \Sref{sec:namespaces}.)
Ralf Jung's avatar
Ralf Jung committed
92

Ralf Jung's avatar
Ralf Jung committed
93
We can further define the notions of \emph{view shifts} and \emph{linear view shifts}:
Ralf Jung's avatar
Ralf Jung committed
94
\begin{align*}
95
  \prop \vsW[\mask_1][\mask_2] \propB \eqdef{}& \prop \wand \pvs[\mask_1][\mask_2] \propB \\
96 97
  \prop \vs[\mask_1][\mask_2] \propB \eqdef{}& \always(\prop \wand \pvs[\mask_1][\mask_2] \propB) \\
  \prop \vs[\mask] \propB \eqdef{}& \prop \vs[\mask][\mask] \propB
Ralf Jung's avatar
Ralf Jung committed
98
\end{align*}
Ralf Jung's avatar
Ralf Jung committed
99
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.
100 101 102 103
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}
104
  {\ownGhost\gname{\melt} \vs[\emptyset] \exists \meltB \in \meltsB.\; \ownGhost\gname{\meltB}}
105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123
\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}}
124
  {\later \prop \vs[\emptyset] \prop}
125

126 127 128 129 130 131 132 133 134 135
% \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 }
%
136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152
\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}

153
\subsection{Weakest Precondition}
Ralf Jung's avatar
Ralf Jung committed
154

155
Finally, we can define the core piece of the program logic, the proposition that reasons about program behavior: Weakest precondition, from which Hoare triples will be derived.
156 157

\paragraph{Defining weakest precondition.}
Ralf Jung's avatar
Ralf Jung committed
158
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 $\Sig$).
159
We further assume (as a parameter) a predicate $\stateinterp : \State \to \iProp$ that interprets the physical state as an Iris proposition.
160
This can be instantiated, for example, with ownership of an authoritative RA to tie the physical state to fragments that are used for user-level proofs.
Ralf Jung's avatar
Ralf Jung committed
161
Finally, weakest precondition takes a parameter $\stuckness \in \set{\NotStuck, \MaybeStuck}$ indicating whether program execution is allowed to get stuck.
Ralf Jung's avatar
Ralf Jung committed
162 163

\begin{align*}
Ralf Jung's avatar
Ralf Jung committed
164
  \textdom{wp}(\stateinterp, \stuckness) \eqdef{}& \MU \textdom{wp\any rec}. \Lam \mask, \expr, \pred. \\
165
        & (\Exists\val. \toval(\expr) = \val \land \pvs[\mask] \pred(\val)) \lor {}\\
166
        & \Bigl(\toval(\expr) = \bot \land \All \state. \stateinterp(\state) \vsW[\mask][\emptyset] {}\\
Ralf Jung's avatar
Ralf Jung committed
167
        &\qquad (s = \NotStuck \Ra \red(\expr, \state)) * \later\All \expr', \state', \vec\expr. (\expr, \state \step \expr', \state', \vec\expr) \vsW[\emptyset][\mask] {}\\
Ralf Jung's avatar
Ralf Jung committed
168
        &\qquad\qquad \stateinterp(\state') * \textdom{wp\any rec}(\mask, \expr', \pred) * \Sep_{\expr'' \in \vec\expr} \textdom{wp\any rec}(\top, \expr'', \Lam \any. \TRUE)\Bigr) \\
Ralf Jung's avatar
Ralf Jung committed
169
%  (* value case *)
Ralf Jung's avatar
Ralf Jung committed
170
  \wpre[\stateinterp]\expr[\stuckness;\mask]{\Ret\val. \prop} \eqdef{}& \textdom{wp}(\stateinterp,\stuckness)(\mask, \expr, \Lam\val.\prop)
Ralf Jung's avatar
Ralf Jung committed
171
\end{align*}
172 173
The $\stateinterp$ will always be set by the context; typically, when instantiating Iris with a language, we also pick the corresponding state interpretation $\stateinterp$.
All proof rules leave $\stateinterp$ unchanged.
Ralf Jung's avatar
Ralf Jung committed
174 175
If we leave away the mask $\mask$, we assume it to default to $\top$.
If we leave away the stuckness $\stuckness$, it defaults to $\NotStuck$.
Ralf Jung's avatar
Ralf Jung committed
176

177
\paragraph{Laws of weakest precondition.}
Ralf Jung's avatar
Ralf Jung committed
178
The following rules can all be derived:
179 180
\begin{mathpar}
\infer[wp-value]
Ralf Jung's avatar
Ralf Jung committed
181
{}{\prop[\val/\var] \proves \wpre{\val}[\stuckness;\mask]{\Ret\var.\prop}}
182 183

\infer[wp-mono]
Ralf Jung's avatar
Ralf Jung committed
184 185
{\mask_1 \subseteq \mask_2 \and \vctx,\var:\textlog{val}\mid\prop \proves \propB \and (\stuckness_2 = \MaybeStuck \lor \stuckness_1 = \stuckness_2)}
{\vctx\mid\wpre\expr[\stuckness_1;\mask_1]{\Ret\var.\prop} \proves \wpre\expr[\stuckness_2;\mask_2]{\Ret\var.\propB}}
186

Ralf Jung's avatar
Ralf Jung committed
187
\infer[fup-wp]
Ralf Jung's avatar
Ralf Jung committed
188
{}{\pvs[\mask] \wpre\expr[\stuckness;\mask]{\Ret\var.\prop} \proves \wpre\expr[\stuckness;\mask]{\Ret\var.\prop}}
189

Ralf Jung's avatar
Ralf Jung committed
190
\infer[wp-fup]
Ralf Jung's avatar
Ralf Jung committed
191
{}{\wpre\expr[\stuckness;\mask]{\Ret\var.\pvs[\stuckness;\mask] \prop} \proves \wpre\expr[\stuckness;\mask]{\Ret\var.\prop}}
192 193

\infer[wp-atomic]
Ralf Jung's avatar
Ralf Jung committed
194 195 196 197
{\stuckness = \NotStuck \Ra \atomic(\expr) \and
 \stuckness = \MaybeStuck \Ra \stronglyAtomic(\expr)}
{\pvs[\mask_1][\mask_2] \wpre\expr[\stuckness;\mask_2]{\Ret\var. \pvs[\mask_2][\mask_1]\prop}
 \proves \wpre\expr[\stuckness;\mask_1]{\Ret\var.\prop}}
198 199

\infer[wp-frame]
Ralf Jung's avatar
Ralf Jung committed
200
{}{\propB * \wpre\expr[\stuckness;\mask]{\Ret\var.\prop} \proves \wpre\expr[\stuckness;\mask]{\Ret\var.\propB*\prop}}
201 202 203

\infer[wp-frame-step]
{\toval(\expr) = \bot \and \mask_2 \subseteq \mask_1}
Ralf Jung's avatar
Ralf Jung committed
204
{\wpre\expr[\stuckness;\mask_2]{\Ret\var.\prop} * \pvs[\mask_1][\mask_2]\later\pvs[\mask_2][\mask_1]\propB \proves \wpre\expr[\stuckness;\mask_1]{\Ret\var.\propB*\prop}}
205 206 207

\infer[wp-bind]
{\text{$\lctx$ is a context}}
Ralf Jung's avatar
Ralf Jung committed
208
{\wpre\expr[\stuckness;\mask]{\Ret\var. \wpre{\lctx(\ofval(\var))}[\stuckness;\mask]{\Ret\varB.\prop}} \proves \wpre{\lctx(\expr)}[\stuckness;\mask]{\Ret\varB.\prop}}
209 210
\end{mathpar}

211
We will also want a rule that connect weakest preconditions to the operational semantics of the language.
212 213
\begin{mathpar}
  \infer[wp-lift-step]
214
  {\toval(\expr_1) = \bot}
215
  { {\begin{inbox} % for some crazy reason, LaTeX is actually sensitive to the space between the "{ {" here and the "} }" below...
216
        ~~\All \state_1. \stateinterp(\state_1) \vsW[\mask][\emptyset] (\stuckness = \NotStuck \Ra \red(\expr_1,\state_1)) * {}\\\qquad~~ \later\All \expr_2, \state_2, \vec\expr.  (\expr_1, \state_1 \step \expr_2, \state_2, \vec\expr)  \vsW[\emptyset][\mask] \Bigl(\stateinterp(\state_2) * \wpre[\stateinterp]{\expr_2}[\stuckness;\mask]{\Ret\var.\prop} * \Sep_{\expr_\f \in \vec\expr} \wpre[\stateinterp]{\expr_\f}[\stuckness;\top]{\Ret\any.\TRUE}\Bigr)  {}\\\proves \wpre[\stateinterp]{\expr_1}[\stuckness;\mask]{\Ret\var.\prop}
217 218 219
      \end{inbox}} }
\end{mathpar}

220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243
% We can further derive some slightly simpler rules for special cases.
% \begin{mathparpagebreakable}
%   \infer[wp-lift-pure-step]
%   {\All \state_1. \red(\expr_1, \state_1) \and
%    \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}}

%   \infer[wp-lift-atomic-step]
%   {\atomic(\expr_1) \and
%    \red(\expr_1, \state_1)}
%   { {\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}
%   \end{inbox}} }

%   \infer[wp-lift-atomic-det-step]
%   {\atomic(\expr_1) \and
%    \red(\expr_1, \state_1) \and
%    \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}}

%   \infer[wp-lift-pure-det-step]
%   {\All \state_1. \red(\expr_1, \state_1) \\
%    \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}}
% \end{mathparpagebreakable}
244 245


Ralf Jung's avatar
Ralf Jung committed
246
\paragraph{Adequacy of weakest precondition.}
247

248 249 250 251
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.

252
\begin{defn}[Adequacy]
Ralf Jung's avatar
Ralf Jung committed
253
  A program $\expr$ in some initial state $\state$ is \emph{adequate} for stuckness  $\stuckness$ and a set $V \subseteq \Val$ of legal return values ($\expr, \state \vDash_\stuckness V$) if for all $\tpool', \state'$ such that $([\expr], \state) \tpstep^\ast (\tpool', \state')$ we have
254
\begin{enumerate}
Ralf Jung's avatar
Ralf Jung committed
255
\item Safety: If $\stuckness = \NotStuck$, then for any $\expr' \in \tpool'$ we have that either $\expr'$ is a
256
  value, or \(\red(\expr'_i,\state')\):
Ralf Jung's avatar
Ralf Jung committed
257
  \[ \stuckness = \NotStuck \Ra \All\expr'\in\tpool'. \toval(\expr') \neq \bot \lor \red(\expr', \state') \]
258 259
  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.
\item Legal return value: If $\tpool'_1$ (the main thread) is a value $\val'$, then $\val' \in V$:
260
  \[ \All \val',\tpool''. \tpool' = [\val'] \dplus \tpool'' \Ra \val' \in V \]
261 262 263
\end{enumerate}
\end{defn}

Ralf Jung's avatar
Ralf Jung committed
264 265 266
To express the adequacy statement for functional correctness, we assume that the signature $\Sig$ adds a predicate $\pred$ to the logic:
\[ \pred : \Val \to \Prop \in \SigFn \]
Furthermore, we assume that the \emph{interpretation} $\Sem\pred$ of $\pred$ reflects some set $V$ of legal return values into the logic (also see \Sref{sec:model}):
267
\[\begin{array}{rMcMl}
268
  \Sem\pred &:& \Sem{\Val\,} \nfn \Sem\Prop \\
269 270 271 272
  \Sem\pred &\eqdef& \Lam \val. \Lam \any. \setComp{n}{v \in V}
\end{array}\]
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:
273
\begin{align*}
Ralf Jung's avatar
Ralf Jung committed
274
 &\All \mask, \expr, \val, \state.
275
 \\&( \TRUE \proves \pvs[\mask] \Exists \stateinterp. \stateinterp(\state) * \wpre[\stateinterp]{\expr}[\stuckness;\mask]{x.\; \pred(x)}) \Ra
Ralf Jung's avatar
Ralf Jung committed
276
 \\&\expr, \state \vDash_\stuckness V
277
\end{align*}
278
Notice that the state invariant $S$ used by the weakest precondition is chosen \emph{after} doing a fancy update, which allows it to depend on the names of ghost variables that are picked in that initial fancy update.
279

Ralf Jung's avatar
Ralf Jung committed
280
\paragraph{Hoare triples.}
Robbert Krebbers's avatar
Robbert Krebbers committed
281
It turns out that weakest precondition is actually quite convenient to work with, in particular when performing these proofs in Coq.
Ralf Jung's avatar
Ralf Jung committed
282 283
Still, for a more traditional presentation, we can easily derive the notion of a Hoare triple:
\[
284
\hoare{\prop}{\expr}{\Ret\val.\propB}[\mask] \eqdef \always{(\prop \wand \wpre{\expr}[\mask]{\Ret\val.\propB})}
Ralf Jung's avatar
Ralf Jung committed
285
\]
286
We assume the state interpretation $\stateinterp$ to be fixed by the context.
Ralf Jung's avatar
Ralf Jung committed
287

Ralf Jung's avatar
Ralf Jung committed
288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
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 \\
Ralf Jung's avatar
Ralf Jung committed
321
   \atomic(\expr)
Ralf Jung's avatar
Ralf Jung committed
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
  }
  {\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's avatar
Ralf Jung committed
353

354
\subsection{Invariant Namespaces}
Ralf Jung's avatar
Ralf Jung committed
355 356
\label{sec:namespaces}

357
In \Sref{sec:invariants}, we defined a proposition $\knowInv\iname\prop$ expressing knowledge (\ie the proposition is persistent) that $\prop$ is maintained as invariant with name $\iname$.
Ralf Jung's avatar
Ralf Jung committed
358 359 360 361 362 363
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.
364
Namespaces are sets of invariants, following a tree-like structure:
Ralf Jung's avatar
Ralf Jung committed
365 366 367 368 369 370 371 372
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's avatar
Robbert Krebbers committed
373
Formally speaking, let $\namesp \in \textlog{InvNamesp} \eqdef \List(\nat)$ be the type of \emph{invariant namespaces}.
Ralf Jung's avatar
Ralf Jung committed
374 375 376 377
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's avatar
Robbert Krebbers committed
378
They, too, are lists of $\nat$, the same type as namespaces.
379 380
In order to connect this up to the definitions of \Sref{sec:invariants}, we need a way to map structued invariant names to $\InvName$, 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 $\InvName$ is infinite.
Ralf Jung's avatar
Ralf Jung committed
381 382
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: \[\namecl\namesp \eqdef \setComp{\iname}{\Exists \namesp'. \iname = \textlog{namesp\_inj}(\namesp' \dplus \namesp)}\]

383
We will overload the notation for invariant propositions for using namespaces instead of names:
Ralf Jung's avatar
Ralf Jung committed
384
\[ \knowInv\namesp\prop \eqdef \Exists \iname \in \namecl\namesp. \knowInv\iname{\prop} \]
Ralf Jung's avatar
Ralf Jung committed
385
We can now derive the following rules (this involves unfolding the definition of fancy updates):
Ralf Jung's avatar
Ralf Jung committed
386 387
\begin{mathpar}
  \axiomH{inv-persist}{\knowInv\namesp\prop \proves \always\knowInv\namesp\prop}
Ralf Jung's avatar
Ralf Jung committed
388

389
  \axiomH{inv-alloc}{\later\prop \proves \pvs[\emptyset] \knowInv\namesp\prop}
Ralf Jung's avatar
Ralf Jung committed
390

Ralf Jung's avatar
Ralf Jung committed
391 392 393
  \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's avatar
Ralf Jung committed
394

Ralf Jung's avatar
Ralf Jung committed
395 396 397 398
  \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's avatar
Ralf Jung committed
399

Ralf Jung's avatar
Ralf Jung committed
400 401 402 403
\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.
404
Accessors are propositions of the form
Ralf Jung's avatar
Ralf Jung committed
405 406
\[ \prop \vs[\mask_1][\mask_2] \Exists\var. \propB * (\All\varB. \propB' \vsW[\mask_2][\mask_1] \propC) \]

407 408
One way to think about such propositions is as follows:
Given some accessor, if during our verification we have the proposition $\prop$ and the mask $\mask_1$ available, we can use the accessor to \emph{access} $\propB$ and obtain the witness $\var$.
Ralf Jung's avatar
Ralf Jung committed
409 410 411 412 413
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 $\propC$.

414 415 416 417 418 419 420 421 422 423
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:
\begin{mathpar}
  \inferH{Acc-vs}
  {\prop \vs[\mask_1][\mask_2] \Exists\var. \propB * (\All\varB. \propB' \vsW[\mask_2][\mask_1] \propC) \and
   \All\var. \propB * \prop_F \vs[\mask_2] \Exists\varB. \propB' * \prop_F}
  {\prop * \prop_F \vs[\mask_1] \propC * \prop_F}

  \inferH{Acc-Ht}
  {\prop \vs[\mask_1][\mask_2] \Exists\var. \propB * (\All\varB. \propB' \vsW[\mask_2][\mask_1] \propC) \and
   \All\var. \hoare{\propB * \prop_F}\expr{\Exists\varB. \propB' * \prop_F}[\mask_2] \and
Ralf Jung's avatar
Ralf Jung committed
424
   \atomic(\expr)}
425 426 427
  {\hoare{\prop * \prop_F}\expr{\propC * \prop_F}[\mask_1]}
\end{mathpar}

Ralf Jung's avatar
Ralf Jung committed
428 429 430 431 432
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.
433

434 435
For the special case that $\prop = \propC$ and $\propB = \propB'$, we use the following notation that avoids repetition:
\[ \Acc[\mask_1][\mask_2]\prop{\Ret x. \propB} \eqdef \prop \vs[\mask_1][\mask_2] \Exists\var. \propB * (\propB \vsW[\mask_2][\mask_1] \prop)  \]
Robbert Krebbers's avatar
Robbert Krebbers committed
436
This accessor is ``idempotent'' in the sense that it does not actually change the state.  After applying it, we get our $\prop$ back so we end up where we started.
437

438 439 440 441
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "iris"
%%% End: