Design and Verification of Adaptive Cache Coherence Protocols ...

that the message queues all become empty in a drained term. Thiscanbeproved according to
Lemma 15, which guarantees that a Cache, WbAck, FlushAck or Wb message can be consumed
at the cache, and a Purge message can be consumed at the memory.
Lemma 20 Given a WP term s,
(1) Pmb id(s) = Pmb id(dr(s))
(2) Mpb id(s) = Mpb id(dr(s))
(3) Proc id(s) = Proc id(dr(s))
Lemma 21 Given a WP term s,
(1) Min(dr(s)) =
(2) Mout(dr(s)) =
(3) Cin id(dr(s)) =
(4) Cout id(dr(s)) =
The draining rules have no impact on the value of a memory cell. Acache cell in a stable state
remains una ected. An uncached address remains uncached provided that no Cache message
is drained. Lemma 22 captures these properties.
Lemma 22 Given a WP term s,
(1) Cell(a,v,T[-,-]) 2 Mem(s) ) Cell(a,v,T[-, ]) 2 Mem(dr(s))
(2) Cell(a,v,Clean) 2 Cache id(s) ) Cell(a,v,Clean) 2 Cache id(dr(s))
(3) Cell(a,v,Dirty) 2 Cache id(s) ) Cell(a,v,Dirty) 2 Cache id (dr(s))
(4) a =2 Cache id(s) ^ Msg(H,id ,Cache,a,-) =2 MoutCin id(s1) ) a =2 Cache id(dr(s))
Lemma 23 ensures that in a drained term, an address will be cached in the Clean state if a
Cache or WbAck message is drained an address will be cached in the Dirty state if a Wb
message is drained an address will be uncached if a Purge or FlushAck message is drained.
The proof follows from Lemma 15 and the draining rules (note that messages can be drained
in any order because of the con uence of the draining rules).
Lemma 23 Given a WP term s,
(1) Msg(H,id ,Cache,a,v) 2 MoutCin id (s) ) Cell(a,v,Clean) 2 Cache id(dr(s))
(2) Msg(id ,H,Purge,a) 2 MinCout id(s) ) a =2 Cache id(dr(s))
(3) Msg(id ,H,Wb,a,v) 2 MinCout id (s) ) Cell(a,v,Dirty) 2 Cache id(dr(s))
(4) Cell(a,v,T[-,(id ,v)j-]) 2 Mem(s) ) Cell(a,v,Dirty) 2 Cache id(dr(s))
(5) Msg(H,id ,WbAck,a) 2 MoutCin id (s) ) Cell(a,v,Clean) 2 Cache id(dr(s))
(6) Msg(H,id ,FlushAck,a) 2 MoutCin id (s) ^ Msg(H,id ,Cache,a,-) =2 MoutCin id(s)
) a =2 Cache id(dr(s))
De nition 24 (Mapping from WP to CRF) Given a WP term s, the corresponding CRF
term f(s) is the drained term dr(s) with all message queues and cache identi ers removed.
It is obvious that the mapping function maps the initial WP term (with all caches and
network queues empty) to the initial CRF term (with all semantic caches empty). For any WP
term, the mapping function guarantees that it is mapped to a legal CRF term (this follows
trivially from the simulation theorem given below).
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