constructions.tex 21.5 KB
Newer Older
1
% !TEX root = ./appendix.tex
Ralf Jung's avatar
Ralf Jung committed
2
\section{COFE constructions}
3

Ralf Jung's avatar
Ralf Jung committed
4
5
6
7
\subsection{Next (type-level later)}

Given a COFE $\cofe$, we define $\latert\cofe$ as follows:
\begin{align*}
8
  \latert\cofe \eqdef{}& \latertinj(x:\cofe) \\
Ralf Jung's avatar
Ralf Jung committed
9
10
  \latertinj(x) \nequiv{n} \latertinj(y) \eqdef{}& n = 0 \lor x \nequiv{n-1} y
\end{align*}
11
12
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's avatar
Ralf Jung committed
13
14
$\latert(-)$ is a locally \emph{contractive} functor from $\COFEs$ to $\COFEs$.

15

Ralf Jung's avatar
Ralf Jung committed
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
\subsection{Uniform Predicates}

Given a CMRA $\monoid$, we define the COFE $\UPred(\monoid)$ of \emph{uniform predicates} over $\monoid$ as follows:
\begin{align*}
  \UPred(\monoid) \eqdef{} \setComp{\pred: \mathbb{N} \times \monoid \to \mProp}{
  \begin{inbox}[c]
    (\All n, x, y. \pred(n, x) \land x \nequiv{n} y \Ra \pred(n, y)) \land {}\\
    (\All n, m, x, y. \pred(n, x) \land x \mincl y \land m \leq n \land y \in \mval_m \Ra \pred(m, y))
  \end{inbox}
}
\end{align*}
where $\mProp$ is the set of meta-level propositions, \eg Coq's \texttt{Prop}.
$\UPred(-)$ is a locally non-expansive functor from $\CMRAs$ to $\COFEs$.

One way to understand this definition is to re-write it a little.
31
We start by defining the COFE of \emph{step-indexed propositions}: For every step-index, the proposition either holds or does not hold.
Ralf Jung's avatar
Ralf Jung committed
32
33
\begin{align*}
  \SProp \eqdef{}& \psetdown{\mathbb{N}} \\
Ralf Jung's avatar
Ralf Jung committed
34
35
    \eqdef{}& \setComp{X \in \pset{\mathbb{N}}}{ \All n, m. n \geq m \Ra n \in X \Ra m \in X } \\
  X \nequiv{n} Y \eqdef{}& \All m \leq n. m \in X \Lra m \in Y
Ralf Jung's avatar
Ralf Jung committed
36
\end{align*}
37
Notice that this notion of $\SProp$ is already hidden in the validity predicate $\mval_n$ of a CMRA:
Ralf Jung's avatar
Ralf Jung committed
38
We could equivalently require every CMRA to define $\mval_{-}(-) : \monoid \nfn \SProp$, replacing \ruleref{cmra-valid-ne} and \ruleref{cmra-valid-mono}.
Ralf Jung's avatar
Ralf Jung committed
39

Ralf Jung's avatar
Ralf Jung committed
40
41
Now we can rewrite $\UPred(\monoid)$ as monotone step-indexed predicates over $\monoid$, where the definition of a ``monotone'' function here is a little funny.
\begin{align*}
Ralf Jung's avatar
Ralf Jung committed
42
  \UPred(\monoid) \cong{}& \monoid \monra \SProp \\
Ralf Jung's avatar
Ralf Jung committed
43
44
45
     \eqdef{}& \setComp{\pred: \monoid \nfn \SProp}{\All n, m, x, y. n \in \pred(x) \land x \mincl y \land m \leq n \land y \in \mval_m \Ra m \in \pred(y)}
\end{align*}
The reason we chose the first definition is that it is easier to work with in Coq.
Ralf Jung's avatar
Ralf Jung committed
46
47

\clearpage
48
\section{RA and CMRA constructions}
49

Ralf Jung's avatar
Ralf Jung committed
50
51
52
\subsection{Product}
\label{sec:prodm}

53
Given a family $(M_i)_{i \in I}$ of CMRAs ($I$ finite), we construct a CMRA for the product $\prod_{i \in I} M_i$ by lifting everything pointwise.
Ralf Jung's avatar
Ralf Jung committed
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74

Frame-preserving updates on the $M_i$ lift to the product:
\begin{mathpar}
  \inferH{prod-update}
  {\melt \mupd_{M_i} \meltsB}
  {f[i \mapsto \melt] \mupd \setComp{ f[i \mapsto \meltB]}{\meltB \in \meltsB}}
\end{mathpar}

\subsection{Finite partial function}
\label{sec:fpfnm}

Given some countable $K$ and some CMRA $\monoid$, the set of finite partial functions $K \fpfn \monoid$ is equipped with a COFE and CMRA structure by lifting everything pointwise.

We obtain the following frame-preserving updates:
\begin{mathpar}
  \inferH{fpfn-alloc-strong}
  {\text{$G$ infinite} \and \melt \in \mval}
  {\emptyset \mupd \setComp{[\gname \mapsto \melt]}{\gname \in G}}

  \inferH{fpfn-alloc}
  {\melt \in \mval}
75
  {\emptyset \mupd \setComp{[\gname \mapsto \melt]}{\gname \in K}}
Ralf Jung's avatar
Ralf Jung committed
76
77
78
79
80

  \inferH{fpfn-update}
  {\melt \mupd \meltsB}
  {f[i \mapsto \melt] \mupd \setComp{ f[i \mapsto \meltB]}{\meltB \in \meltsB}}
\end{mathpar}
81
82
Remember that $\mval$ is the set of elements of a CMRA that are valid at \emph{all} step-indices.

Ralf Jung's avatar
Ralf Jung committed
83
$K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.
Ralf Jung's avatar
Ralf Jung committed
84

85
86
\subsection{Agreement}

Ralf Jung's avatar
Ralf Jung committed
87
Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:
Ralf Jung's avatar
Ralf Jung committed
88
89
\newcommand{\aginjc}{\mathrm{c}} % the "c" field of an agreement element
\newcommand{\aginjV}{\mathrm{V}} % the "V" field of an agreement element
Ralf Jung's avatar
Ralf Jung committed
90
\begin{align*}
Ralf Jung's avatar
Ralf Jung committed
91
  \agm(\cofe) \eqdef{}& \record{\aginjc : \mathbb{N} \to \cofe , \aginjV : \SProp} \\
Ralf Jung's avatar
Ralf Jung committed
92
  & \text{quotiented by} \\
Ralf Jung's avatar
Ralf Jung committed
93
94
  \melt \equiv \meltB \eqdef{}& \melt.\aginjV = \meltB.\aginjV \land \All n. n \in \melt.\aginjV \Ra \melt.\aginjc(n) \nequiv{n} \meltB.\aginjc(n) \\
  \melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.\aginjV \Lra m \in \meltB.\aginjV) \land (\All m \leq n. m \in \melt.\aginjV \Ra \melt.\aginjc(m) \nequiv{m} \meltB.\aginjc(m)) \\
95
  \mval_n \eqdef{}& \setComp{\melt \in \agm(\cofe)}{ n \in \melt.\aginjV \land \All m \leq n. \melt.\aginjc(n) \nequiv{m} \melt.\aginjc(m) } \\
Ralf Jung's avatar
Ralf Jung committed
96
  \mcore\melt \eqdef{}& \melt \\
Ralf Jung's avatar
Ralf Jung committed
97
  \melt \mtimes \meltB \eqdef{}& (\melt.\aginjc, \setComp{n}{n \in \melt.\aginjV \land n \in \meltB.\aginjV \land \melt \nequiv{n} \meltB })
Ralf Jung's avatar
Ralf Jung committed
98
\end{align*}
99
100
Note that the carrier $\agm(\cofe)$ is a \emph{record} consisting of the two fields $\aginjc$ and $\aginjV$.

Ralf Jung's avatar
Ralf Jung committed
101
$\agm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$.
Ralf Jung's avatar
Ralf Jung committed
102

Ralf Jung's avatar
Ralf Jung committed
103
You can think of the $\aginjc$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in \aginjV$ steps.
104
The reason we store a chain, rather than a single element, is that $\agm(\cofe)$ needs to be a COFE itself, so we need to be able to give a limit for every chain of $\agm(\cofe)$.
Ralf Jung's avatar
Ralf Jung committed
105
However, given such a chain, we cannot constructively define its limit: Clearly, the $\aginjV$ of the limit is the limit of the $\aginjV$ of the chain.
106
But what to pick for the actual data, for the element of $\cofe$?
Ralf Jung's avatar
Ralf Jung committed
107
Only if $\aginjV = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $\aginjV$ is smaller, the chain ``cancels'', \ie stops converging as we reach indices $n \notin \aginjV$.
108
To mitigate this, we apply the usual construction to close a set; we go from elements of $\cofe$ to chains of $\cofe$.
Ralf Jung's avatar
Ralf Jung committed
109

Ralf Jung's avatar
Ralf Jung committed
110
111
We define an injection $\aginj$ into $\agm(\cofe)$ as follows:
\[ \aginj(x) \eqdef \record{\mathrm c \eqdef \Lam \any. x, \mathrm V \eqdef \mathbb{N}} \]
Ralf Jung's avatar
Ralf Jung committed
112
113
There are no interesting frame-preserving updates for $\agm(\cofe)$, but we can show the following:
\begin{mathpar}
Ralf Jung's avatar
Ralf Jung committed
114
  \axiomH{ag-val}{\aginj(x) \in \mval_n}
115

Ralf Jung's avatar
Ralf Jung committed
116
  \axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)}
117
  
Ralf Jung's avatar
Ralf Jung committed
118
  \axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Ra x \nequiv{n} y}
Ralf Jung's avatar
Ralf Jung committed
119
120
\end{mathpar}

Ralf Jung's avatar
Ralf Jung committed
121
122
123
124
125
\subsection{One-shot}

The purpose of the one-shot CMRA is to lazily initialize the state of a ghost location.
Given some CMRA $\monoid$, we define $\oneshotm(\monoid)$ as follows:
\begin{align*}
Ralf Jung's avatar
Ralf Jung committed
126
  \oneshotm(\monoid) \eqdef{}& \ospending + \osshot(\monoid) + \munit + \bot \\
Ralf Jung's avatar
Ralf Jung committed
127
  \mval_n \eqdef{}& \set{\ospending, \munit} \cup \setComp{\osshot(\melt)}{\melt \in \mval_n}
Ralf Jung's avatar
Ralf Jung committed
128
129
\\%\end{align*}
%\begin{align*}
Ralf Jung's avatar
Ralf Jung committed
130
131
132
  \osshot(\melt) \mtimes \osshot(\meltB) \eqdef{}& \osshot(\melt \mtimes \meltB) \\
  \munit \mtimes \ospending \eqdef{}& \ospending \mtimes \munit \eqdef \ospending \\
  \munit \mtimes \osshot(\melt) \eqdef{}& \osshot(\melt) \mtimes \munit \eqdef \osshot(\melt)
Ralf Jung's avatar
Ralf Jung committed
133
\end{align*}%
134
Notice that $\oneshotm(\monoid)$ is a disjoint sum with the four constructors (injections) $\ospending$, $\osshot$, $\munit$ and $\bot$.
Ralf Jung's avatar
Ralf Jung committed
135
The remaining cases of composition go to $\bot$.
Ralf Jung's avatar
Ralf Jung committed
136
137
138
139
\begin{align*}
  \mcore{\ospending} \eqdef{}& \munit & \mcore{\osshot(\melt)} \eqdef{}& \mcore\melt \\
  \mcore{\munit} \eqdef{}& \munit &  \mcore{\bot} \eqdef{}& \bot
\end{align*}
Ralf Jung's avatar
Ralf Jung committed
140
141
142
143
144
145
146
147
148
149
The step-indexed equivalence is inductively defined as follows:
\begin{mathpar}
  \axiom{\ospending \nequiv{n} \ospending}

  \infer{\melt \nequiv{n} \meltB}{\osshot(\melt) \nequiv{n} \osshot(\meltB)}

  \axiom{\munit \nequiv{n} \munit}

  \axiom{\bot \nequiv{n} \bot}
\end{mathpar}
Ralf Jung's avatar
Ralf Jung committed
150
$\oneshotm(-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.
Ralf Jung's avatar
Ralf Jung committed
151

Ralf Jung's avatar
Ralf Jung committed
152
153
154
155
156
157
158
159
160
161
We obtain the following frame-preserving updates:
\begin{mathpar}
  \inferH{oneshot-shoot}
  {\melt \in \mval}
  {\ospending \mupd \osshot(\melt)}

  \inferH{oneshot-update}
  {\melt \mupd \meltsB}
  {\osshot(\melt) \mupd \setComp{\osshot(\meltB)}{\meltB \in \meltsB}}
\end{mathpar}
162

Ralf Jung's avatar
Ralf Jung committed
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
\subsection{Exclusive CMRA}

Given a cofe $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned:
\begin{align*}
  \exm(\cofe) \eqdef{}& \exinj(\cofe) + \munit + \bot \\
  \mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \bot} \\
  \munit \mtimes \exinj(x) \eqdef{}& \exinj(x) \mtimes \munit \eqdef \exinj(x)
\end{align*}
The remaining cases of composition go to $\bot$.
\begin{align*}
  \mcore{\exinj(x)} \eqdef{}& \munit & \mcore{\munit} \eqdef{}& \munit &
  \mcore{\bot} \eqdef{}& \bot
\end{align*}
The step-indexed equivalence is inductively defined as follows:
\begin{mathpar}
  \infer{x \nequiv{n} y}{\exinj(x) \nequiv{n} \exinj(y)}
179

Ralf Jung's avatar
Ralf Jung committed
180
  \axiom{\munit \nequiv{n} \munit}
181

Ralf Jung's avatar
Ralf Jung committed
182
183
184
185
186
187
188
189
190
191
192
193
194
  \axiom{\bot \nequiv{n} \bot}
\end{mathpar}
$\exm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$.

We obtain the following frame-preserving update:
\begin{mathpar}
  \inferH{ex-update}{}
  {\exinj(x) \mupd \exinj(y)}
\end{mathpar}



%TODO: These need syncing with Coq
195
196
197
198
199
200
201
202
203
204
205
206
207
208
% \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's avatar
Ralf Jung committed
209
% 	Assume some frame $\melt_\f \sep \emptyset$. Since $\melt_\f$ is finite and $X$ is infinite, there exists an $x \notin \melt_\f$.
210
211
212
213
214
% 	Pick that for the result.
% \end{proof}

% The powerset monoids is cancellative.
% \begin{proof}[Proof of cancellativity]
Ralf Jung's avatar
Ralf Jung committed
215
216
217
218
% 	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 $x \in meltB$.
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
245
246
247
248
249
250
% 	The other direction works the same way.
% \end{proof}


% \subsection{Fractional monoid}
% \label{sec:fracm}

% Given a monoid $M$, we define a monoid representing fractional ownership of some piece $\melt \in M$.
% The idea is to preserve all the frame-preserving update that $M$ could have, while additionally being able to do \emph{any} update if we own the full state (as determined by the fraction being $1$).
% Let $\fracm{M}$ be the monoid with carrier $(((0, 1] \cap \mathbb{Q}) \times M) \uplus \{\munit\}$ and multiplication
% \begin{align*}
%  (q, a) \mtimes (q', a') &\eqdef (q + q', a \mtimes a') \qquad \mbox{if $q+q'\le 1$} \\
%  (q, a) \mtimes \munit &\eqdef (q,a) \\
%  \munit \mtimes (q,a) &\eqdef (q,a).
% \end{align*}

% We get the following frame-preserving update.
% \begin{mathpar}
% 	\inferH{FracUpdFull}
% 		{a, b \in M}
% 		{(1, a) \mupd (1, b)}
%   \and\inferH{FracUpdLocal}
% 	  {a \mupd_M B}
% 	  {(q, a) \mupd \{q\} \times B}
% \end{mathpar}

% \begin{proof}[Proof of \ruleref{FracUpdFull}]
% Assume some $f \sep (1, a)$. This can only be $f = \munit$, so showing $f \sep (1, b)$ is trivial.
% \end{proof}

% \begin{proof}[Proof of \ruleref{FracUpdLocal}]
% 	Assume some $f \sep (q, a)$. If $f = \munit$, then $f \sep (q, b)$ is trivial for any $b \in B$. Just pick the one we obtain by choosing $\munit_M$ as the frame for $a$.
251
	
Ralf Jung's avatar
Ralf Jung committed
252
253
% 	In the interesting case, we have $f = (q_\f, a_\f)$.
% 	Obtain $b$ such that $b \in B \land b \sep a_\f$.
254
255
256
257
258
% 	Then $(q, b) \sep f$, and we are done.
% \end{proof}

% $\fracm{M}$ is cancellative if $M$ is cancellative.
% \begin{proof}[Proof of cancellativitiy]
Ralf Jung's avatar
Ralf Jung committed
259
260
% If $\melt_\f = \munit$, we are trivially done.
% So let $\melt_\f = (q_\f, \melt_\f')$.
261
262
263
264
% If $\melt = \munit$, then $\meltB = \munit$ as otherwise the fractions could not match up.
% Again, we are trivially done.
% Similar so for $\meltB = \munit$.
% So let $\melt = (q_a, \melt')$ and $\meltB = (q_b, \meltB')$.
Ralf Jung's avatar
Ralf Jung committed
265
% We have $(q_\f + q_a, \melt_\f' \mtimes \melt') = (q_\f + q_b, \melt_\f' \mtimes \meltB')$.
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
% We have to show $q_a = q_b$ and $\melt' = \meltB'$.
% The first is trivial, the second follows from cancellativitiy of $M$.
% \end{proof}


% %\subsection{Disposable monoid}
% %
% %Given a monoid $M$, we construct a monoid where, having full ownership of an element $\melt$ of $M$, one can throw it away, transitioning to a dead element.
% %Let \dispm{M} be the monoid with carrier $\mcarp{M} \uplus \{ \disposed \}$ and multiplication
% %% The previous unit must remain the unit of the new monoid, as is is always duplicable and hence we could not transition to \disposed if it were not composable with \disposed
% %\begin{align*}
% %  \melt \mtimes \meltB &\eqdef \melt \mtimes_M \meltB & \IF \melt \sep[M] \meltB \\
% %  \disposed \mtimes \disposed &\eqdef \disposed \\
% %  \munit_M \mtimes \disposed &\eqdef \disposed \mtimes \munit_M \eqdef \disposed
% %\end{align*}
% %The unit is the same as in $M$.
% %
% %The frame-preserving updates are
% %\begin{mathpar}
% % \inferH{DispUpd}
% %   {a \in \mcarp{M} \setminus \{\munit_M\} \and a \mupd_M B}
% %   {a \mupd B}
% % \and
% % \inferH{Dispose}
% %  {a \in \mcarp{M} \setminus \{\munit_M\} \and \All b \in \mcarp{M}. a \sep b \Ra b = \munit_M}
% %  {a \mupd \disposed}
% %\end{mathpar}
% %
% %\begin{proof}[Proof of \ruleref{DispUpd}]
% %Assume a frame $f$. If $f = \disposed$, then $a = \munit_M$, which is a contradiction.
% %Thus $f \in \mcarp{M}$ and we can use $a \mupd_M B$.
% %\end{proof}
% %
% %\begin{proof}[Proof of \ruleref{Dispose}]
% %The second premiss says that $a$ has no non-trivial frame in $M$. To show the update, assume a frame $f$ in $\dispm{M}$. Like above, we get $f \in \mcarp{M}$, and thus $f = \munit_M$. But $\disposed \sep \munit_M$ is trivial, so we are done.
% %\end{proof}

% \subsection{Authoritative monoid}\label{sec:auth}

% Given a monoid $M$, we construct a monoid modeling someone owning an \emph{authoritative} element $x$ of $M$, and others potentially owning fragments $\melt \le_M x$ of $x$.
% (If $M$ is an exclusive monoid, the construction is very similar to a half-ownership monoid with two asymmetric halves.)
% Let $\auth{M}$ be the monoid with carrier
% \[
% 	\setComp{ (x, \melt) }{ x \in \mcarp{\exm{\mcarp{M}}} \land \melt \in \mcarp{M} \land (x = \munit_{\exm{\mcarp{M}}} \lor \melt \leq_M x) }
% \]
% and multiplication
% \[
% (x, \melt) \mtimes (y, \meltB) \eqdef
%      (x \mtimes y, \melt \mtimes \meltB) \quad \mbox{if } x \sep y \land \melt \sep \meltB \land (x \mtimes y = \munit_{\exm{\mcarp{M}}} \lor \melt \mtimes \meltB \leq_M x \mtimes y)
% \]
% Note that $(\munit_{\exm{\mcarp{M}}}, \munit_M)$ is the unit and asserts no ownership whatsoever, but $(\munit_{M}, \munit_M)$ asserts that the authoritative element is $\munit_M$.

% Let $x, \melt \in \mcarp M$.
% We write $\authfull x$ for full ownership $(x, \munit_M):\auth{M}$ and $\authfrag \melt$ for fragmental ownership $(\munit_{\exm{\mcarp{M}}}, \melt)$ and $\authfull x , \authfrag \melt$ for combined ownership $(x, \melt)$.
% If $x$ or $a$ is $\mzero_{M}$, then the sugar denotes $\mzero_{\auth{M}}$.

% \ralf{This needs syncing with the Coq development.}
% The frame-preserving update involves a rather unwieldy side-condition:
% \begin{mathpar}
% 	\inferH{AuthUpd}{
Ralf Jung's avatar
Ralf Jung committed
326
% 		\All\melt_\f\in\mcar{\monoid}. \melt\sep\meltB \land \melt\mtimes\melt_\f \le \meltB\mtimes\melt_\f \Ra \melt'\mtimes\melt_\f \le \melt'\mtimes\meltB \and
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
% 		\melt' \sep \meltB
% 	}{
% 		\authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \melt' \mtimes \meltB, \authfrag \melt'
% 	}
% \end{mathpar}
% We therefore derive two special cases.

% \paragraph{Local frame-preserving updates.}

% \newcommand\authupd{f}%
% Following~\cite{scsl}, we say that $\authupd: \mcar{M} \ra \mcar{M}$ is \emph{local} if
% \[
% 	\All a, b \in \mcar{M}. a \sep b \land \authupd(a) \neq \mzero \Ra \authupd(a \mtimes b) = \authupd(a) \mtimes b
% \]
% Then,
% \begin{mathpar}
% 	\inferH{AuthUpdLocal}
% 	{\text{$\authupd$ local} \and \authupd(\melt)\sep\meltB}
% 	{\authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \authupd(\melt) \mtimes \meltB, \authfrag \authupd(\melt)}
% \end{mathpar}

% \paragraph{Frame-preserving updates on cancellative monoids.}

% Frame-preserving updates are also possible if we assume $M$ cancellative:
% \begin{mathpar}
%  \inferH{AuthUpdCancel}
%   {\text{$M$ cancellative} \and \melt'\sep\meltB}
%   {\authfull \melt \mtimes \meltB, \authfrag \melt \mupd \authfull \melt' \mtimes \meltB, \authfrag \melt'}
% \end{mathpar}

% \subsection{Fractional heap monoid}
% \label{sec:fheapm}

% By combining the fractional, finite partial function, and authoritative monoids, we construct two flavors of heaps with fractional permissions and mention their important frame-preserving updates.
% Hereinafter, we assume the set $\textdom{Val}$ of values is countable.

% Given a set $Y$, define $\FHeap(Y) \eqdef \textdom{Val} \fpfn \fracm(Y)$ representing a fractional heap with codomain $Y$.
% From \S\S\ref{sec:fracm} and~\ref{sec:fpfunm} we obtain the following frame-preserving updates as well as the fact that $\FHeap(Y)$ is cancellative.
% \begin{mathpar}
% 	\axiomH{FHeapUpd}{h[x \mapsto (1, y)] \mupd h[x \mapsto (1, y')]} \and
% 	\axiomH{FHeapAlloc}{h \mupd \{\, h[x \mapsto (1, y)] \mid x \in \textdom{Val} \,\}}
% \end{mathpar}
% We will write $qh$ with $h : \textsort{Val} \fpfn Y$ for the function in $\FHeap(Y)$ mapping every $x \in \dom(h)$ to $(q, h(x))$, and everything else to $\munit$.

% Define $\AFHeap(Y) \eqdef \auth{\FHeap(Y)}$ representing an authoritative fractional heap with codomain $Y$.
% We easily obtain the following frame-preserving updates.
% \begin{mathpar}
% 	\axiomH{AFHeapUpd}{
% 		(\authfull h[x \mapsto (1, y)], \authfrag [x \mapsto (1, y)]) \mupd (\authfull h[x \mapsto (1, y')], \authfrag [x \mapsto (1, y')])
% 	}
% 	\and
% 	\inferH{AFHeapAdd}{
% 		x \notin \dom(h)
% 	}{
% 		\authfull h \mupd (\authfull h[x \mapsto (q, y)], \authfrag [x \mapsto (q, y)])
% 	}
% 	\and
% 	\axiomH{AFHeapRemove}{
% 		(\authfull h[x \mapsto (q, y)], \authfrag [x \mapsto (q, y)]) \mupd \authfull h
% 	}
% \end{mathpar}

389
390
\subsection{STS with tokens}
\label{sec:stsmon}
391

392
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 a monoid modeling an authoritative current state and permitting transitions given a \emph{bound} on the current state and a set of \emph{locally-owned} tokens.
393

394
395
396
397
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's avatar
Ralf Jung committed
398
 s \stsfstep{T} s' \eqdef{}& \Exists T_1, T_2. T_1 \disj \STSL(s) \cup T \l+and (s, T_1) \stsstep (s', T_2)
399
\end{align*}
400

401
402
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's avatar
Ralf Jung committed
403
\STSclsd(S, T) \eqdef{}& \All s \in S. \STSL(s) \disj T \land \All s'. s \stsfstep{T} s' \Ra s' \in S \\
404
405
\upclose(S, T) \eqdef{}& \setComp{ s' \in \STSS}{\Exists s \in S. s \stsftrans{T} s' }
\end{align*}
406

407
408
The STS RA is defined as follows
\begin{align*}
Ralf Jung's avatar
Ralf Jung committed
409
410
411
  \monoid \eqdef{}& \setComp{\STSauth((s, T) \in \STSS \times \wp(\STST))}{\STSL(s) \disj T} +{}\\& \setComp{\STSfrag((S, T) \in \wp(\STSS) \times \wp(\STST))}{\STSclsd(S, T) \land S \neq \emptyset} + \bot \\
  \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 $s \in S$} \\
412
413
414
415
  \mcore{\STSfrag(S, T)} \eqdef{}& \STSfrag(\upclose(S, \emptyset), \emptyset) \\
  \mcore{\STSauth(s, T)} \eqdef{}& \STSfrag(\upclose(\set{s}, \emptyset), \emptyset)
\end{align*}
The remaining cases are all $\bot$.
416

417
418
419
420
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')}
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
448
449
450
451
452
453
  \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})}
454

455
456
457
458
459
460
461
  {\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}
462
463
464
465
466

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "iris"
%%% End: