Design and Verification of Adaptive Cache Coherence Protocols ...

s1 if p (s1)
! s2
where s1 and s2 are terms, and p is a predicate.
We can use a rule to rewrite a term if the rule's left-hand-side pattern matches the term
or one of its subterms and the corresponding predicate is true. The new term is generated in
accordance with the right-hand-side of the rule. If several rules apply, then any one of them
can be applied. If no rule applies, then the term cannot be rewritten any further and is said
to be in normal form. In practice, we often use abstract data types such as arrays and FIFO
queues to make the descriptions more readable.
As an example, the Euclides algorithm for computing the greatest common divisor of two
numbers can be expressed as follows:
GCD(x,y) if x < y ! GCD(y,x)
GCD(x,y) if x y ^y 6= 0 ! GCD(x-y,y)
The SK combinatory system, which has only two rules and a simple grammar for generating
terms, provides a small but fascinating example of term rewriting. These two rules are su cient
to describe any computable function.
K-rule: (K.x).y ! x
S-rule: ((S.x).y).z ! (x.z).(y.z)
We canverify that, for any subtermx, the term ((S.K).K).x can be rewritten to (K.x).(K.x)
by applying the S-rule. We can rewrite the term further to x by applying the K-rule. Thus,
if we read the dot as a function application, then the term ((S.K).K) behaves as the identity
function. Notice the S-rule rearranges the dot and duplicates the term represented by x on the
right-hand-side. For architectures in which terms represent states, rules must be restricted so
that terms are not restructured or duplicated as in the S and K rules.
We say term s1 can be rewritten to term s2 in zero or more steps (s1 ! s2) ifs1 and s2 are
identical or there exists a term s 0 such that s1 can be rewritten to s 0 in one step (s1 ! s 0 )and
s 0 can be rewritten to s2 in zero or more steps (s 0 ! s2). A TRS is con uent ifforanyterms1,
if s1 ! s2 and s1 ! s3, then there exists aterms4 such that s2 ! s4 and s3 ! s4. ATRSis
strongly terminating if a term can always be rewritten to a normal form regardless of the order
in which the rules are applied. More information about TRSs can be found elsewhere [14, 69].
2.2 The AX Instruction Set
We use AX, a minimalist RISC instruction set, to illustrate all the processor examples in this
chapter. The TRS description of a simple AX architecture also provides a good introductory
example to the TRS notation. In the AX instruction set (Figure 2.1), all arithmetic operations
27