Design and Verification of Adaptive Cache Coherence Protocols ...

Proof The proof is based on a case analysis on the imperative ruleusedin the rewriting of
`s1 ! s2' in Migratory. Let a be the address and id the cache identi er.
Imperative Processor Rules
If Rule IP1 (Loadl-on-Clean) applies, then
Cell(a,v,Clean) 2 Cache id(s1) ^ Cell(a,v,Clean) 2 Cache id(s2)
) Cell(a,v,C[ ]) 2 Mem(s1) ^ Cell(a,v,C[ ]) 2 Mem(s2) (Lemma 41 )
) f(s1) ! f(s2) (CRF-Cache, CRF-Loadl & CRF-Purge )
If Rule IP2 (Loadl-on-Dirty) applies, then
Cell(a,v,Dirty) 2 Cache id(s1) ^ Cell(a,v,Dirty) 2 Cache id(s2)
) Cell(a,v,C[ ]) 2 Mem(s1) ^ Cell(a,v,C[ ]) 2 Mem(s2) (Lemma 41 )
) f(s1) ! f(s2) (CRF-Cache, CRF-Loadl & CRF-Purge )
If Rule IP3 (Storel-on-Clean) applies, then
Cell(a,-,Clean) 2 Cache id(s1) ^ Cell(a,v,Dirty) 2 Cache id(s2)
) Cell(a,-,C[ ]) 2 Mem(s1) ^ Cell(a,v,C[ ]) 2 Mem(s2) (Lemma 41 )
) f(s1) ! f(s2) (CRF-Cache, CRF-Storel, CRF-Writeback & CRF-Purge )
If Rule IP4 (Storel-on-Dirty) applies, then
Cell(a,-,Dirty) 2 Cache id(s1) ^ Cell(a,v,Dirty) 2 Cache id(s2)
) Cell(a,-,C[ ]) 2 Mem(s1) ^ Cell(a,v,C[ ]) 2 Mem(s2) (Lemma 41 )
) f(s1) ! f(s2) (CRF-Cache, CRF-Storel, CRF-Writeback & CRF-Purge )
If Rule IP5 (Commit-on-Clean), IP6 (Commit-on-Dirty) orIP7(Commit-on-Invalid) applies, then
f(s1) ! f(s2) (CRF-Commit )
If Rule IP8 (Reconcile-on-Clean), IP9 (Reconcile-on-Dirty) or IP10 (Reconcile-on-Invalid) applies,
then
f(s1) ! f(s2) (CRF-Reconcile )
Imperative M-engine Rules
If Rule IM1 (M-Send-Cache) applies, then
Draining Rules
f(s1) =f(s2)
If Rule IC1 (C-Send-Purge), IC2 (C-Send-Flush), IC3 (C-Receive-Cache), IM1 (M-Receive-Purge),
IM2 (M-Receive-Flush), Message-Cache-to-Mem or Message-Mem-to-Cache applies, then
f(s1) =f(s2) (Since the rule or its backward version is a draining rule)
Figure 6.6 summarizes the simulation proof. 2
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