constructions.tex 17.6 KB
Newer Older
Ralf Jung's avatar
Ralf Jung committed
1
\section{COFE constructions}
2

Ralf Jung's avatar
Ralf Jung committed
3
4
\subsection{Next (type-level later)}

5
Given a COFE $\cofe$, we define $\latert\cofe$ as follows (using a datatype-like notation to define the type):
Ralf Jung's avatar
Ralf Jung committed
6
\begin{align*}
7
  \latert\cofe \eqdef{}& \latertinj(x:\cofe) \\
Ralf Jung's avatar
Ralf Jung committed
8
9
  \latertinj(x) \nequiv{n} \latertinj(y) \eqdef{}& n = 0 \lor x \nequiv{n-1} y
\end{align*}
10
11
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
12
13
$\latert(-)$ is a locally \emph{contractive} functor from $\COFEs$ to $\COFEs$.

14

Ralf Jung's avatar
Ralf Jung committed
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
\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.
30
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
31
32
\begin{align*}
  \SProp \eqdef{}& \psetdown{\mathbb{N}} \\
Ralf Jung's avatar
Ralf Jung committed
33
34
    \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
35
\end{align*}
36
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
37
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
38

Ralf Jung's avatar
Ralf Jung committed
39
40
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
41
  \UPred(\monoid) \cong{}& \monoid \monra \SProp \\
Ralf Jung's avatar
Ralf Jung committed
42
43
44
     \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
45
46

\clearpage
47
\section{RA and CMRA constructions}
48

Ralf Jung's avatar
Ralf Jung committed
49
50
51
\subsection{Product}
\label{sec:prodm}

52
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
53
54
55
56
57
58
59
60

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}

61
62
63
\subsection{Sum}
\label{sec:summ}

64
The \emph{sum CMRA} $\monoid_1 \csumm \monoid_2$ for any CMRAs $\monoid_1$ and $\monoid_2$ is defined as (again, we use a datatype-like notation):
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
\begin{align*}
  \monoid_1 \csumm \monoid_2 \eqdef{}& \cinl(\melt_1:\monoid_1) \mid \cinr(\melt_2:\monoid_2) \mid \bot \\
  \mval_n \eqdef{}& \setComp{\cinl(\melt_1)\!}{\!\melt_1 \in \mval'_n}
    \cup \setComp{\cinr(\melt_2)\!}{\!\melt_2 \in \mval''_n}  \\
  \cinl(\melt_1) \mtimes \cinl(\meltB_1) \eqdef{}& \cinl(\melt_1 \mtimes \meltB_1)  \\
%  \munit \mtimes \ospending \eqdef{}& \ospending \mtimes \munit \eqdef \ospending \\
%  \munit \mtimes \osshot(\melt) \eqdef{}& \osshot(\melt) \mtimes \munit \eqdef \osshot(\melt) \\
  \mcore{\cinl(\melt_1)} \eqdef{}& \begin{cases}\mnocore & \text{if $\mcore{\melt_1} = \mnocore$} \\ \cinl({\mcore{\melt_1}}) & \text{otherwise} \end{cases}
\end{align*}
The composition and core for $\cinr$ are defined symmetrically.
The remaining cases of the composition and core are all $\bot$.
Above, $\mval'$ refers to the validity of $\monoid_1$, and $\mval''$ to the validity of $\monoid_2$.

We obtain the following frame-preserving updates, as well as their symmetric counterparts:
\begin{mathpar}
  \inferH{sum-update}
  {\melt \mupd_{M_1} \meltsB}
  {\cinl(\melt) \mupd \setComp{ \cinl(\meltB)}{\meltB \in \meltsB}}

  \inferH{sum-swap}
  {\All \melt_\f, n. \melt \mtimes \melt_\f \notin \mval'_n \and \meltB \in \mval''}
  {\cinl(\melt) \mupd \cinr(\meltB)}
\end{mathpar}
Crucially, the second rule allows us to \emph{swap} the ``side'' of the sum that the CMRA is on if $\mval$ has \emph{no possible frame}.

Ralf Jung's avatar
Ralf Jung committed
90
91
92
\subsection{Finite partial function}
\label{sec:fpfnm}

Ralf Jung's avatar
Ralf Jung committed
93
Given some infinite 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.
Ralf Jung's avatar
Ralf Jung committed
94
95
96
97
98
99
100
101
102

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}
103
  {\emptyset \mupd \setComp{[\gname \mapsto \melt]}{\gname \in K}}
Ralf Jung's avatar
Ralf Jung committed
104
105

  \inferH{fpfn-update}
106
  {\melt \mupd_\monoid \meltsB}
Ralf Jung's avatar
Ralf Jung committed
107
108
  {f[i \mapsto \melt] \mupd \setComp{ f[i \mapsto \meltB]}{\meltB \in \meltsB}}
\end{mathpar}
109
Above, $\mval$ refers to the validity of $\monoid$.
110

Ralf Jung's avatar
Ralf Jung committed
111
$K \fpfn (-)$ is a locally non-expansive functor from $\CMRAs$ to $\CMRAs$.
Ralf Jung's avatar
Ralf Jung committed
112

113
114
\subsection{Agreement}

Ralf Jung's avatar
Ralf Jung committed
115
116
Given some COFE $\cofe$, we define $\agm(\cofe)$ as follows:
\begin{align*}
117
118
119
120
121
122
  \agm(\cofe) \eqdef{}& \set{(c, V) \in (\mathbb{N} \to \cofe) \times \SProp}/\ {\sim} \\[-0.2em]
  \textnormal{where }& \melt \sim \meltB \eqdef{} \melt.V = \meltB.V \land 
    \All n. n \in \melt.V \Ra \melt.c(n) \nequiv{n} \meltB.c(n)  \\
%    \All n \in {\melt.V}.\, \melt.x \nequiv{n} \meltB.x \\
  \melt \nequiv{n} \meltB \eqdef{}& (\All m \leq n. m \in \melt.V \Lra m \in \meltB.V) \land (\All m \leq n. m \in \melt.V \Ra \melt.c(m) \nequiv{m} \meltB.c(m)) \\
  \mval_n \eqdef{}& \setComp{\melt \in \agm(\cofe)}{ n \in \melt.V \land \All m \leq n. \melt.c(n) \nequiv{m} \melt.c(m) } \\
Ralf Jung's avatar
Ralf Jung committed
123
  \mcore\melt \eqdef{}& \melt \\
124
  \melt \mtimes \meltB \eqdef{}& \left(\melt.c, \setComp{n}{n \in \melt.V \land n \in \meltB.V \land \melt \nequiv{n} \meltB }\right)
Ralf Jung's avatar
Ralf Jung committed
125
\end{align*}
126
%Note that the carrier $\agm(\cofe)$ is a \emph{record} consisting of the two fields $c$ and $V$.
127

Ralf Jung's avatar
Ralf Jung committed
128
$\agm(-)$ is a locally non-expansive functor from $\COFEs$ to $\CMRAs$.
Ralf Jung's avatar
Ralf Jung committed
129

130
You can think of the $c$ as a \emph{chain} of elements of $\cofe$ that has to converge only for $n \in V$ steps.
131
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 $\agm(\cofe)$.
132
However, given such a chain, we cannot constructively define its limit: Clearly, the $V$ of the limit is the limit of the $V$ of the chain.
133
But what to pick for the actual data, for the element of $\cofe$?
134
Only if $V = \mathbb{N}$ we have a chain of $\cofe$ that we can take a limit of; if the $V$ is smaller, the chain ``cancels'', \ie stops converging as we reach indices $n \notin V$.
135
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
136

Ralf Jung's avatar
Ralf Jung committed
137
138
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
139
140
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
141
  \axiomH{ag-val}{\aginj(x) \in \mval_n}
142

Ralf Jung's avatar
Ralf Jung committed
143
  \axiomH{ag-dup}{\aginj(x) = \aginj(x)\mtimes\aginj(x)}
144
  
Ralf Jung's avatar
Ralf Jung committed
145
  \axiomH{ag-agree}{\aginj(x) \mtimes \aginj(y) \in \mval_n \Ra x \nequiv{n} y}
Ralf Jung's avatar
Ralf Jung committed
146
147
\end{mathpar}

148

Ralf Jung's avatar
Ralf Jung committed
149
150
\subsection{Exclusive CMRA}

Ralf Jung's avatar
Ralf Jung committed
151
Given a COFE $\cofe$, we define a CMRA $\exm(\cofe)$ such that at most one $x \in \cofe$ can be owned:
Ralf Jung's avatar
Ralf Jung committed
152
\begin{align*}
153
154
  \exm(\cofe) \eqdef{}& \exinj(\cofe) + \bot \\
  \mval_n \eqdef{}& \setComp{\melt\in\exm(\cofe)}{\melt \neq \bot}
Ralf Jung's avatar
Ralf Jung committed
155
\end{align*}
156
All cases of composition go to $\bot$.
Ralf Jung's avatar
Ralf Jung committed
157
\begin{align*}
158
  \mcore{\exinj(x)} \eqdef{}& \mnocore &
Ralf Jung's avatar
Ralf Jung committed
159
160
  \mcore{\bot} \eqdef{}& \bot
\end{align*}
161
162
Remember that $\mnocore$ is the ``dummy'' element in $\maybe\monoid$ indicating (in this case) that $\exinj(x)$ has no core.

Ralf Jung's avatar
Ralf Jung committed
163
164
165
The step-indexed equivalence is inductively defined as follows:
\begin{mathpar}
  \infer{x \nequiv{n} y}{\exinj(x) \nequiv{n} \exinj(y)}
166

Ralf Jung's avatar
Ralf Jung committed
167
168
169
170
171
172
173
174
175
176
177
178
179
  \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
180
181
182
183
184
185
186
187
188
189
190
191
192
193
% \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
194
% 	Assume some frame $\melt_\f \sep \emptyset$. Since $\melt_\f$ is finite and $X$ is infinite, there exists an $x \notin \melt_\f$.
195
196
197
198
199
% 	Pick that for the result.
% \end{proof}

% The powerset monoids is cancellative.
% \begin{proof}[Proof of cancellativity]
Ralf Jung's avatar
Ralf Jung committed
200
201
202
203
% 	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 $x \in meltB$.
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
% 	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 $a$.
236
	
Ralf Jung's avatar
Ralf Jung committed
237
238
% 	In the interesting case, we have $f = (q_\f, a_\f)$.
% 	Obtain $b$ such that $b \in B \land b \sep a_\f$.
239
240
241
242
243
% 	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
244
245
% If $\melt_\f = \munit$, we are trivially done.
% So let $\melt_\f = (q_\f, \melt_\f')$.
246
247
248
249
% 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
250
% We have $(q_\f + q_a, \melt_\f' \mtimes \melt') = (q_\f + q_b, \melt_\f' \mtimes \meltB')$.
251
252
253
254
255
% We have to show $q_a = q_b$ and $\melt' = \meltB'$.
% The first is trivial, the second follows from cancellativitiy of $M$.
% \end{proof}


Ralf Jung's avatar
Ralf Jung committed
256
257
\subsection{Authoritative}
\label{sec:auth-cmra}
258

Ralf Jung's avatar
Ralf Jung committed
259
Given a CMRA $M$, we construct $\authm(M)$ modeling someone owning an \emph{authoritative} element $\melt$ of $M$, and others potentially owning fragments $\meltB \mincl \melt$ of $\melt$.
Ralf Jung's avatar
Ralf Jung committed
260
261
262
263
264
265
266
267
268
269
We assume that $M$ has a unit $\munit$, and hence its core is total.
(If $M$ is an exclusive monoid, the construction is very similar to a half-ownership monoid with two asymmetric halves.)
\begin{align*}
\authm(M) \eqdef{}& \maybe{\exm(M)} \times M \\
\mval_n \eqdef{}& \setComp{ (x, \meltB) \in \authm(M) }{ \meltB \in \mval_n \land (x = \mnocore \lor \Exists \melt. x = \exinj(\melt) \land \meltB \mincl_n \melt) } \\
  (x_1, \meltB_1) \mtimes (x_2, \meltB_2) \eqdef{}& (x_1 \mtimes x_2, \meltB_2 \mtimes \meltB_2) \\
  \mcore{(x, \meltB)} \eqdef{}& (\mnocore, \mcore\meltB) \\
  (x_1, \meltB_1) \nequiv{n} (x_2, \meltB_2) \eqdef{}& x_1 \nequiv{n} x_2 \land \meltB_1 \nequiv{n} \meltB_2
\end{align*}
Note that $(\mnocore, \munit)$ is the unit and asserts no ownership whatsoever, but $(\exinj(\munit), \munit)$ asserts that the authoritative element is $\munit$.
270

Ralf Jung's avatar
Ralf Jung committed
271
272
Let $\melt, \meltB \in M$.
We write $\authfull \melt$ for full ownership $(\exinj(\melt), \munit)$ and $\authfrag \meltB$ for fragmental ownership $(\mnocore, \meltB)$ and $\authfull \melt , \authfrag \meltB$ for combined ownership $(\exinj(\melt), \meltB)$.
273

Ralf Jung's avatar
Ralf Jung committed
274
275
276
277
278
279
280
The frame-preserving update involves the notion of a \emph{local update}:
\newcommand\lupd{\stackrel{\mathrm l}{\mupd}}
\begin{defn}
  It is possible to do a \emph{local update} from $\melt_1$ and $\meltB_1$ to $\melt_2$ and $\meltB_2$, written $(\melt_1, \meltB_1) \lupd (\melt_2, \meltB_2)$, if
  \[ \All n, \maybe{\melt_\f}. x_1 \in \mval_n \land \melt_1 \nequiv{n} \meltB_1 \mtimes \maybe{\melt_\f} \Ra \melt_2 \in \mval_n \land \melt_2 \nequiv{n} \meltB_2 \mtimes \maybe{\melt_\f} \]
\end{defn}
In other words, the idea is that for every possible frame $\maybe{\melt_\f}$ completing $\meltB_1$ to $\melt_1$, the same frame also completes $\meltB_2$ to $\melt_2$.
281

Ralf Jung's avatar
Ralf Jung committed
282
283
284
285
286
287
We then obtain
\begin{mathpar}
  \inferH{auth-update}
  {(\melt_1, \meltB_1) \lupd (\melt_2, \meltB_2)}
  {\authfull \melt_1 , \authfrag \meltB_1 \mupd \authfull \melt_2 , \authfrag \meltB_2}
\end{mathpar}
288

289
\subsection{STS with tokens}
Ralf Jung's avatar
Ralf Jung committed
290
\label{sec:sts-cmra}
291

Ralf Jung's avatar
Ralf Jung committed
292
Given a state-transition system~(STS, \ie a directed graph) $(\STSS, {\stsstep} \subseteq \STSS \times \STSS)$, a set of tokens $\STST$, and a labeling $\STSL: \STSS \ra \wp(\STST)$ of \emph{protocol-owned} tokens for each state, we construct an RA modeling an authoritative current state and permitting transitions given a \emph{bound} on the current state and a set of \emph{locally-owned} tokens.
293

294
295
296
297
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
298
 s \stsfstep{T} s' \eqdef{}& \Exists T_1, T_2. T_1 \disj \STSL(s) \cup T \land (s, T_1) \stsstep (s', T_2)
299
\end{align*}
300

301
302
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
303
\STSclsd(S, T) \eqdef{}& \All s \in S. \STSL(s) \disj T \land \left(\All s'. s \stsfstep{T} s' \Ra s' \in S\right) \\
304
305
\upclose(S, T) \eqdef{}& \setComp{ s' \in \STSS}{\Exists s \in S. s \stsftrans{T} s' }
\end{align*}
306

307
308
The STS RA is defined as follows
\begin{align*}
Ralf Jung's avatar
Ralf Jung committed
309
310
311
  \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 $s \in S$} \\
312
313
314
315
  \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 $\bot$.
316

317
318
319
320
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')}
321

322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
  \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})}
354

355
356
357
358
359
360
361
  {\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}
362
363
364
365
366

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