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Revised for model of 03/28/07.
Contents |
Dekker
Consider the program.
p=1 p=2 [1] x=1 [3] y=1 [1s]sync [3s] sync [2] _=y(0) [4] _=x(0)
By program order and (F1):
- [1] <1 [1s], [1s] <1 [2], [3] <2 [3s], [3s] <2 [4]
Now <1 totally orders E(1) u L; hence it fixes the order between [2] and [4]. Suppose [2] <1 [4]. Then by Load Completeness, [2] <2 [4]. Thus [2] in V(2), and hence by HW1, [1] in V(2), and in fact [1] <2 [2] <2 [4]. Therefore [4] must return 1 and this execution is not feasible. Suppose now [4] <1 [2]. By similar reasoning, HW1 forces [3] <1 [4] <1 [2], and hence [2] must return 1, and this execution is not feasible.
The execution is also ruled out by PPCq, for the following reason. On processor 1, the load [4] must be ordered before the store [1] because they conflict and must therefore be ordered coherently (PPCq requires that within each processor's timeline, conflicting accesses are ordered coherently if at least one of them is local). Thus, [4] <1 [1]. But we (as detailed above) we also know [1] <1 [2] and [2] <1 [3], and [3] <1 [4], so we get a cycle in the order <1 contradicting the requirement that <1 be a total order.
A limus test program can be found here.
IRIW --Independent Read of Independent Write
Consider the program:
p=1 p=2 p=3 p=4 [1] _=x(1) [3] _=y(1) [5] x=1 [6] y=1 [1s]sync [3s] sync [2] _=y(0) [4] _=x(0)
The application of A-cumulativity (HW1) reduces this to Dekker, hence the Dekker analysis establishes that this execution is not possible.
The PPCq allows this execution, as the following processor timelines satisfy all axioms:
- for processor 1 we have [5] <1 [1] <1 [1s] <1 [2] <1 [6] <1 [3] <1 [3s] < [4]
- for processor 2 we have [6] <2 [3] <2 [3s] <2 [4] <2 [5] <2 [1] <2 [1s] < [2]
- for processor 3, 4 we can pick the same timeline as for processor 2.
PPCq allows [5] <1 [4] (even though it is not coherent) because both [5] and [4] are remote to processor 1. It only requires coherent order between two accesses if (1) at least one of them is a write, and (2) at least one of them is local.
A limus test program can be found here.
CC -- Causal Consistency
Consider the program:
p=1 p=2 p=3 [1a] _=x(1) [3] _=y(1) [1] x=1 [1s]sync [3s] sync [2] y=1 [4] _=x(0)
This is not an execution of PPCw, hence not an execution in PPCw or PPCs).
(V2) forces [1] to be visible at p=1, and [1] <1 [1a], and hence [1] <1 [2]. (F1) causes [1] <2 [2]. (V2) forces [2] <2 [3]. Therefore it follows that [1] <2 [4] and hence [4] must return 1.
DCl -- Direct Consistency
Consider the program:
p=1 p=2 p=3 [1a] _=x(1) [3] y=1 [1] x=1 [1s]sync [3s] sync [2] _=y(0) [4] _=x(0)
Again an application of (HW1) reduces this to Dekker.
LCC --Causal Consistency with Long chain
This example is due to Sebastian Burckhardt. Consider the program:
p=1 p=2 p=3 p=4 [1a] _=x(1) [3] _=y(1) [1] x=1 [5] _=z(1) [1s]sync [3s] sync [5s] sync [2] y=1 [4] z=1 [6] _=x(0)
The same reasoning as in CC, carried through one more step.
(V2) forces [1] to be visible at p=1, and [1] <1 [1a], and hence [1] <1 [2]. (HW1) causes [1] <2 [2]. (V2) forces [2] <2 [3]. Therefore it follows that [1] <2 [3s] <2 [4]. Now (F1) can be applied to transfer [1] <2 [4] to <4. This forces [6] to read 1.
LF -- Lock-free stack/queue/list
This example is due to Maged Michael.
p=1 p=2 [1] x=1 [3] _=y(1) [1s] sync [3s] sync [2] y=1 [4] r2=x(0)
This example is a direct application of (F2) to [1],[2],[3] and [4] yielding [1] <2 [4], hence [4] must return 1 in all three models.
SN --Snapshot
This example is due to Maged Michael.
p=1 p=2 p=3 p=4 [1a] _=x(1) [3] _=x(0) [1] x=1 [3] y=1 [1s] sync [3s] sync [2] _=y(0) [4] _=y(1) [2s] sync [4s] sync [2b] _=x(1) [4b] _=x(0)
This is not permitted by PPCl. Either [2] <1 [4] or [4] <1 [2].
In the first case, by Load Completeness, [2] <2 [4], and hence by (HW1) [1] <2 [4], and hence [4b] will return 1, contradicting the desired result.
In the second case, by (HW1) [3] <1 [4] <1 [2], hence [2] must return 1.
