Design and Verification of Adaptive Cache Coherence Protocols ...

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Figure 2.12: Combination of Forward and Backward Draining
Show that the draining rules are strongly terminating and con uent, that is, rewriting a
term with respect to the draining rules always terminates and reachs the same normal
form, regardless of the order in which the rules are applied. This ensures the existence of
a unique drained term for each implementation term.
De ne a mapping function that maps an implementation term to a speci cation term.
The mapping function can often be speci ed as a projection function that removes un-
necessary states from drained terms. Show that the mapping function maps the initial
implementation term to the initial speci cation term.
Prove that the speci cation can simulate the implementation with respect to the mapping
function. That is, if s1 can be rewritten to s2 in the implementation, then the correspond-
ing term of s1 can be rewritten to the corresponding term of s2 in the speci cation.
2.5.2 Veri cation of Liveness
We use temporal logic to reason about time-varying behaviors and liveness properties. An
execution of a system can be described as a sequence of rewriting steps, each producing a new
term by applying a rewriting rule on the current term. A sequence is a sequence of terms hs1,
s2, s3, :::i where s1 is a legal term (that is, s0 ! s1 where s0 is the initial term) and si ! si+1
(for i =1,2,:::).
A predicate can be de ned using boolean operators and the temporal operator \2" (always).
It can be a predicate for terms whichcontains no temporal operator, or a predicate for sequences
of terms which contains some temporal operators. We say a sequence satis es a term predicate
if the rst term of the sequence satis es the predicate, and a sequence satis es a rewriting rule
if the rst term of the sequence can be rewritten to the second term of the sequence according
to the rule. Since all the boolean operators can be de ned in terms of \:" and \^", it su ces
to de ne the semantics of predicates as follows:
(:P )( ) :P ( )
(P ^Q )( ) P ( ) ^ Q ( )
(2P )( ) 8i 2f1 2:::g P (hsi, si+1, si+2, :::i)
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