Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

2.2. CCS, FORMALLY 33
• If B is empty, then it is only willing to accept one datum as input along a
channel called ‘in’. The received datum is stored for further output.
• If B is full, then it is only willing to output the successor of the value it stores,
and empties itself in doing so.
This behaviour of B can be modelled in value passing CCS thus:
B(x)
B def
= in(x).B(x)
def
= out(x + 1).B .
Note that the input prefix ‘in’ now carries a parameter that is a variable—in this
case x—whose scope is the process that is prefixed by the input action—in this
example, B(x). The intuitive idea is that process B is willing to accept a nonnegative
integer n as input, bind the received value to x and thereafter behave like
B(n)—that is, like a full one-place buffer storing the value n. The behaviour of
the process B(n) is then described by the second equation above, where the scope
of the formal parameter x is the whole right-hand side of the equation. Note that
output prefixes, like ‘out(x+1)’ above, may carry expressions—the idea being that
the value being output is the one that results from the evaluation of the expression.
The general SOS rule for input prefixing now becomes
a(x).P a(n)
→ P [n/x]
n ≥ 0
where we write P [n/x] for the expression that results by replacing each free occurrence
of the variable x in P with n. The general SOS rule for output prefixing
is instead the one below.
ā(e).P ā(n)
→ P
n is the result of evaluating e
In value passing CCS, as we have already seen in our definition of the one place
buffer B, process names may be parameterized by value variables. The general
form that these parameterized constants may take is A(x1, . . . , xn), where A is a
process name, n ≥ 0 and x1, . . . , xn are distinct value variables. The operational
semantics for these constants is given by the following rule.
P [v1/x1, . . . , vn/xn] α → P ′
A(e1, . . . , en) α → P ′
A(x1, . . . , xn) def
= P and each ei has value vi