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

Chapter 4
Theory of fixed points and
bisimulation equivalence
The aim of this chapter is to collect under one roof all the mathematical notions
from the theory of partially ordered sets and lattices that is needed to introduce
Tarski’s classic fixed point theorem. You might think that this detour into some exotic
looking mathematics is unwarranted in this textbook. However, we shall then
put these possible doubts of yours to rest by using this fixed point theorem to give
an alternative definition of strong bisimulation equivalence. This reformulation of
the notion of strong bisimulation equivalence is not just mathematically pleasing,
but it also yields an algorithm for computing the largest strong bisimulation over finite
labelled transition systems—i.e., labelled transition systems with only finitely
many states, actions and transitions. This is an illustrative example of how apparently
very abstract mathematical notions turn out to have algorithmic content and,
possibly unexpected, applications in Computer Science. As you will see in what
follows, we shall also put Tarski’s fixed point theorem to good use in Chapter 6,
where the theory developed in this chapter will allow us to understand the meaning
of recursively defined properties of reactive systems.
4.1 Posets and complete lattices
We start our technical developments in this chapter by introducing the notion of
partially ordered set (also known as poset) and some useful classes of such structures
that will find application in what follows. As you will see, you are already
familiar with many of the examples of posets that we shall mention in this chapter.
Definition 4.1 [Partially ordered sets] A partially ordered set (often abbreviated to
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