The C11 and C++11 Concurrency Model

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in composition with a client will be a subset of that of the other. From this observation,
we define a sound abstraction relation over histories that we can then lift to the sets of
histories generated by library code.
With this formulation of abstraction, a specification is simply a collection of histories.
Here, our implementation and specification are both programs, and we will enumerate
the histories of each by executing them in an arbitrary client context. This motivates the
definition of the most general client: rather than enumerate the behaviour of the library
in an arbitrary client context, we would like a constrained set of client contexts that are
sufficient to generate all possible histories. The most general client must enumerate all
possible combinations of library calls, on any number of threads, with all possible values
of arguments. The definition of the most general client is:
Definition 18. The most general client is defined as follows: Take n ≥ 1 and let
{m 1 ,...,m l } be the methods implemented by a library L. We let
MGC n (L) = (let L in C mgc
1 ‖ ... ‖ C mgc
n ),
where C mgc
t
is
while(nondet()) { if(nondet()) {m 1 }else if(nondet()) {m 2 } ... else {m l } }
Here, we let the parameters of library methods be chosen arbitrarily.
Weareconsideringarestrictedsetofprogramswherealllibrarylocationsareinitialised
with writes that happen before all other memory accesses. We write LI for the set of
executions of the library L under the most general client starting from an initial state
I.
To cover the set of all possible consistent executions, we must also enumerate all
possible client happens-before edges: the presence of a client happens-before edge does
not simply restrict the set of consistent executions; it can introduce new ones, by creating
a new visible-side-effect for a non-atomic read, allowing a new value to be read, for
example. We define the extension of an execution X with the relation R as an execution
with identical components whose happens-before relation is transitively extended with R.
Now we can extend the execution of the most general client with an arbitrary set of client
happens-before edges so that we capture all possible behaviours of the library. We write
L,RI for the set of consistent executions of L from I extended with R.
In Lemma 22 we establish the following: all library projections of an execution of a
client and a library are contained in the execution of the library under the most general
client, extended with client happens-before edges. This shows that the MGC can
reproduce the behaviour of the library under any client, and that it is, in fact, most
general.