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

96 CHAPTER 4. THEORY OF FIXED POINTS
4.3 Bisimulation as a fixed point
Now that we have the theory underlying Tarski’s fixed point theorem in place,
it is high time to put it into practice. We shall first use the theory we have just
developed to provide the promised reformulation of bisimulation equivalence, and
next we shall show by means of examples how this reformulation leads directly to
an algorithm for computing bisimilarity over finite labelled transition systems. The
algorithm for computing bisimilarity that stems from the theory of fixed points is
not the most efficient one that has been devised; however, it is really pleasing
to see how apparently very abstract notions from mathematics turn out to have
unexpected applications in computer science.
Throughout this section, we let (Proc, Act, { α → | α ∈ Act}) be a labelled transition
system. We recall that a relation R ⊆ Proc×Proc is a strong bisimulation—
see Definition 3.2 on page 43—if the following holds:
If (p, q) ∈ R then, for every α ∈ Act:
1. p α → p ′ implies q α → q ′ for some q ′ such that (p ′ , q ′ ) ∈ R;
2. q α → q ′ implies p α → p ′ for some p ′ such that (p ′ , q ′ ) ∈ R.
Then strong bisimulation equivalence (or strong bisimilarity) is defined as
∼ = {R ∈ 2 (Proc×Proc) | R is a strong bisimulation} .
In what follows we shall describe the relation ∼ as a fixed point to a suitable monotonic
function. First we note that (2 (Proc×Proc) , ⊆) (i.e., the set of binary relations
over Proc ordered by set inclusion) is a complete lattice with and as
least upper bound and greatest lower bound. (Why? In fact, you should be able
to realize readily that we have seen this kind of complete lattice in our previous
developments!)
Consider now a binary relation R over Proc—that is, an element of the set
2 (Proc×Proc) . We define the set F(R) as follows:
(p, q) ∈ F(R), for all p, q ∈ Proc, if and only if
1. p α → p ′ implies q α → q ′ for some q ′ such that (p ′ , q ′ ) ∈ R;
2. q α → q ′ implies p α → p ′ for some p ′ such that (p ′ , q ′ ) ∈ R.
In other words, F(R) contains all the pairs of processes from which, in one round
of the bisimulation game, the defender can make sure that the players reach a current
pair of processes that is already contained in R.