Commentary on PPCL Model
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What is the model intended to do? It is intended to capture in as precise a fashion as possible -- using as basic mathematics as possible -- the rules that govern correct execution for the PPC (as far as programs containing just loads, stores and hwsyncs are concerned).
A goal of the formalization is to stay as close as possible to the language in the Redbooks.
So how does the model work. Basically, it says that an execution of a program consists of a set of events (load, store and fence) events, and some relationships between them. In particular there is a per processor total order (sequence) of events (one for each processor, <p), and a per variable total order of write events to that variable (<w). Then there are some conditions that relate these.
What does <p capture? It captures the relationship between events (loads, stores) being performed at a processor, together with the events arising from instructions being executed by the processor. It helps to separate out remote events from local events. This is done formally by saying that there is a function, p, which for every event tells you the processor that is executing the event. So in the context of a given process q, a local event is an event e such that p(e)=q, and a remote event is an event e such that p(e) != q.
So what is a local load event and how does it relate to the exeution of a load instruction by a processor? A local load event corresponds to the completion of a load instruction, that is the event associated with loading the return value into a register. A local store event, on the other hand, corresponds to getting an index for the write and initiating the communication of the store to other processors. A remote store event corresponds to performing the store at the processor, and a remote load event corresponds to performing the store at the processor. There are no remote fence operations, only local. A fence event corresponds to the execution of the fence instruction.
Note that like any model this model is extremely abstract. The discussion in the previous paragraph should not be thought as specifying any operational view of events, rather a logical view that is detailed enough to correctly specify whether an execution is valid for PPC or not.
Let us continue. What is V(p)? Well, we do not make the assumption that every event in the base set E must be performed with respect to every processor in the system. Indeed, there is explicit language in the Architecture handbooks that says that some writes may never make it to storage and hence may not be visible to other processors. So V(p) is the set of all events which are either executed by the processor p or are performed wrt to p.
Now we come to the first condition (Coherence). It simply says that every processor will observe stores (for any given address a) in the same sequence in which these are written into the variable.
Now another idea baked into the model is in order to determine the value returned by a load l from x performed by a processor p, simply look at all the stores to x that occur in the processor order (<p) before l. If there are none, then return the initial value. Otherwise take the value associated with the last store. All this is captured formally by (V1) and (V2).
(Note: V3 has been removed from the model.) Now what is (V3). It captures the following meaning of a load being performed at a processor. It places the weakest possible restriction (that is the w in PPCw), namely: for a load from x to be performed at a processor p it must be the case that the store whose value is being returned has already been performed at processor p. Note that this does not imply that no fresher stores to x have been performed -- just that any subsequent load executed by p will return a value at least as fresh as that being returned by l.
Now come the treatment of fences. This is basically straightforward and follows straight from the books. The set A corresponds to all events e such that e <p f, where f is the fence we are looking at. Note that we do not require e be local (if it were so, we would have required p(e)=p). This is to handle the idea of cumulativity in the set A. So F1 simply says that for all events e before a fence and for any local events e' after the fence it must be the case that for every processor q, e is performed at q before e' is performed at q. (F2) takes care of B-cumulativity, almost exactly as a translation of the informal language in the redbook.
Now we come to the central idea that lends power to the PPCl model, namely the Completeness Assumption. This assumption introduces a partial order on loads. Consider two loads k and l, with processor p=p(k) and q=p(l). Suppose the execution of k is completed on p, and then l is performed on p. This permits us to conclude that k is performed on q before l. (l completes on q after it is performed on all processors, including p, but it is performed on p after k completes, and k completes after it has performed on all processors, including q. Therefore, k is performed on q before l completes on q.)
This is the fundamental piece of reasoning used to rule out Dekker, and hence IRIW. Vj 11:18, 18 February 2007 (PST)
