Design and Verification of Adaptive Cache Coherence Protocols ...

According to Theorem-C and Lemma 45,
Cell(a,-,T[id ]) 2 Mem( ) ^ Msg(H,id ,FlushReq,a)# 2 Mout( )
(Cell(a,-,T[id ]) 2 Mem( ) ^ Msg(H,id ,FlushReq,a)# 2 Cin id( )) _
Cell(a,-,C[ ]) 2 Mem( )
According to Theorem-C and Lemma 46,
Cell(a,-,T[id ]) 2 Mem( ) ^ Msg(H,id ,FlushReq,a)# 2 Cin id( )
(Cell(a,-,T[id ]) 2 Mem( ) ^ Msg(H,id ,FlushReq,a) l 2 Cin id( )) _
Cell(a,-,C[id ]) 2 Mem( )
According to Lemmas 44, 45, and 46,
Cell(a,-,T[id ]) 2 Mem( ) ^ Msg(H,id ,FlushReq,a) l 2 Cin id( )
Msg(id ,H,Purge,a) 2 MinCout id( ) _ Msg(id ,H,Flush,a,-) 2 MinCout id ( )
Msg(id ,H,Purge,a) 2 Min( ) _ Msg(id ,H,Flush,a,-) 2 Min( )
Msg(id ,H,Purge,a)" 2 Min( ) _ Msg(id ,H,Flush,a,-)" 2 Min( )
Cell(a,-,C[ ]) 2 Mem( )
Thus, Cell(a,-,T[id ]) 2 Mem( ) ^ Msg(H,id ,FlushReq,a)# 2 Mout( )
Cell(a,-,C[ ]) 2 Mem( )
This completes the proof according to Theorem-A. 2
The proof above can be described in a more intuitive way. The memory sends a FlushReq
message to cache site id each timeitchanges the memory state to T[id ]. When the FlushReq
message is received at the cache, if the cache state is Clean or Dirty, the cache sends a Purge
or Flush message to the memory otherwise the cache ignores the FlushReq message. In the
latter case, if the memory state remains as T[id ], there is a Purge or Flush message in transit
from the cache to the memory according to Lemma 44 (note that the memory cannot send any
message to the cache while the memory state is T[id ], and FIFO message passing guarantees
that any preceding message has been received before the Purge or Flush message is received).
When the memory receives the Purge or Flush message, it sets the memory state to C[ ].
Lemma 48 is a general form of Lemma 46. It ensures that an incoming message will eventually
become a message that has no preceding message. Note that Lemma 47 ensures that a stalled
CacheReq message will be processed eventually, since Rule MM1 is strongly fair.
Lemma 48 Given a Migratory sequence ,
(1) msg 2 Cin id( ) msg" 2 Cin id( )
(2) msg 2 Min( ) msg" 2 Min( )
Lemma 49 ensures that if an address is cached in the CachePending state, the cache state will
become Clean eventually.
Lemma 49 Given a Migratory sequence ,
Cell(a,-,CachePending) 2 Cache id( ) Cell(a,-,Clean) 2 Cache id( )
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