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

28 CHAPTER 2. THE LANGUAGE CCS
we have so far developed. However, intuition alone can lead us to wrong conclusions,
and most importantly cannot be fed to a computer! To capture formally
our understanding of the semantics of the language CCS, we therefore introduce
the collection of SOS rules in Table 2.2. These rules are used to generate an LTS
whose states are CCS expressions. In that LTS, a transition P α → Q holds for CCS
expressions P, Q and action α if, and only if, it can be proven using the rules in
Table 2.2.
A rule like
α.P α → P
is an axiom, as it has no premises—that is, it has no transition above the solid
line. This means that proving that a process of the form α.P affords the transition
α.P α → P (the conclusion of the rule) can be done without establishing any further
sub-goal. Therefore each process of the form α.P affords the transition α.P α → P .
As an example, we have that the following transition
pub.CS1
pub
→ CS1
is provable using the above rule for action prefixing.
On the other hand, a rule like
P α → P ′
K α → P ′
def
K = P
(2.7)
has a non-empty set of premises. This rule says that to establish that constant K
affords the transition mentioned in the conclusion of the rule, we have to prove
first that the body of the defining equation for K, namely the process P , affords
the transition P α → P ′ . Using this rule, pattern matching and transition (2.7), we
can prove the transition
CS pub
→ CS1 ,
which we had informally derived before for the version of process CS given in
Table 2.1 on page 18.
The aforementioned rule for constants has a side condition, namely K def
= P ,
that describes a constraint that must be met in order for the rule to be applicable. In
that specific example, the side condition states intuitively that the rule may be used
to derive an initial transition for constant K if ‘K is declared to have body P ’.
Another example of a rule with a side condition is that for restriction.
P α → P ′
P \ L α → P ′ \ L
α, ¯α ∈ L