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

6.6. CHARACTERISTIC PROPERTIES 141
pn
p
a
a
pn−1
a
pn−2 · · · p2
a
p1
Figure 6.3: The processes p and pn
a
p0
Recall that, by the modal depth of a formula F , notation md(F ), we mean the
maximum number of nested occurrences of the modal operators in F . Formally
this is defined by the following recursive definition:
1. md(tt) = md(ff) = 0,
2. md([a]F ) = md(〈a〉F ) = 1 + md(F ),
3. md(F1 ∨ F2) = md(F1 ∧ F2) = max{md(F1), md(F2)}.
Next we define a sequence p0, p1, p2, . . . of processes inductively as follows:
1. p0 = 0,
2. pi+1 = a.pi.
(The processes p and pi, for i ≥ 1, are depicted in Figure 6.3.) Observe that each
process pi can perform a sequence of i a-labelled transitions in a row and terminate
in doing so. Moreover, this is the only behaviour that pi affords.
Now we can prove the following:
p |= F implies p md(F ) |= F, for each F . (6.13)
The statement in (6.13) can be proven by structural induction on F and is left as an
exercise for the reader. As obviously p and pn are not bisimulation equivalent for
any n (why?), the statement in (6.13) contradicts (6.12). Indeed, (6.12) and (6.13)
imply that p is bisimilar to pk, where k is the modal depth of the formula Fp.
As (6.12) is a consequence of (6.11), we can therefore conclude that no recursion
free formula Fp can characterize the process p up to bisimulation equivalence.
Exercise 6.11 Prove statement (6.13).