4 - Theory of fixed points and bisimulation equivalence

Summary

The aim of this chapter is to collect under one roof all the mathematical notions from the theory of partially ordered sets and lattices 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 also yields an algorithm for computing the largest strong bisimulation over finite labelled transition systems (LTSs), 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, 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.

Posets and complete lattices

We start our technical developments in this chapter by introducing the notion of a partially ordered set (also known as a poset) and some useful classes of such structures that will find application in what follows. As you will see, many examples of posets that we shall mention in this chapter are familiar.