We introduce bisimulation and coinduction roughly following the way that led to their discovery in Computer Science. Thus the general topic is the semantics of concurrent languages (or systems), in which several activities, the processes, may run concurrently. Central questions are: what is, mathematically, a process? And what does it mean that two processes are ‘equal’? We seek notions of process and process equality that are both mathematically and practically interesting. For instance, the notions should be amenable to effective techniques for proving equalities, and the equalities themselves should be justifiable, according to the way processes are used.

We hope that the reader will find this way of proceeding helpful for understanding the meaning of bisimulation and coinduction. The emphasis on processes is also justified by the fact that concurrency remains today the main application area for bisimulation and coinduction.

We compare processes and functions in Section 1.1. We will see that processes do not fit the input/output schema of functions. A process has an interactive behaviour, and it is essential to take this into account. We formalise the idea of behaviour in Section 1.2 via labelled transition systems (LTSs), together with notations and terminology for them. We discuss the issue of equality between behaviours in Section 1.3. We first try to re-use notions of equality from Graph Theory and Automata Theory. The failure of these attempts leads us to proposing bisimilarity, in Section 1.4. We introduce the reader to the bisimulation proof method through a number of examples.

This book is about bisimulation and coinduction. It is the companion book of the volume An Introduction to Bisimulation and Coinduction, by Davide Sangiorgi (Cambridge University Press, 2011), which deals with the basics of bisimulation and coinduction, with an emphasis on labelled transition systems, processes, and other notions from the theory of concurrency.

In the present volume, we have collected a number of chapters, by different authors, on several advanced topics in bisimulation and coinduction. These chapters either treat specific aspects of bisimulation and coinduction in great detail, including their history, algorithmics, enhanced proof methods and logic. Or they generalise the basic notions of bisimulation and coinduction to different or more general settings, such as coalgebra, higher-order languages and probabilistic systems. Below we briefly summarise the chapters in this volume.

• The origins of bisimulation and coinduction, by Davide Sangiorgi

In this chapter, the origins of the notions of bisimulation and coinduction are traced back to different fields, notably computer science, modal logic, and set theory.

• An introduction to (co)algebra and (co)induction, by Bart Jacobs and Jan Rutten

Here the notions of bisimulation and coinduction are explained in terms of coalgebras. These mathematical structures generalise all kinds of infinitedata structures and automata, including streams (infinite lists), deterministic and probabilistic automata, and labelled transition systems. Coalgebras are formally dual to algebras and it is this duality that is used to put both induction and coinduction into a common perspective.

This book is an introduction to bisimulation and coinduction and a precursor to the companion book on more advanced topics. Between them, the books analyse the most fundamental aspects of bisimulation and coinduction, exploring concepts and techniques that can be transported to many areas. Bisimulation is a special case of coinduction, by far the most studied coinductive concept. Bisimulation was discovered in Concurrency Theory and processes remain the main application area. This explains the special emphasis on bisimulation and processes that one finds throughout the two volumes.

This volume treats basic topics. It explains coinduction, and its duality with induction, from various angles, starting from some simple results of fixed-point theory. It then goes on to bisimulation, as a tool for defining behavioural equality among processes (bisimilarity), and for proving such equalities. It compares bisimulation with other notions of behavioural equivalence. It also presents a simple process calculus, both to show algebraic techniques for bisimulation and to illustrate the combination of inductive and coinductive reasoning.

The companion volume, Advanced Topics in Bisimulation and Coinduction, edited by Davide Sangiorgi and Jan Rutten, deals with more specialised topics. A chapter recalls the history of the discovery of bisimulation and coinduction. Another chapter unravels the duality between induction and coinduction, both as defining principles and as proof techniques, in terms of the duality between the mathematical notions of algebra and coalgebra and properties such as initiality and finality.

The simulation equivalence of Exercise 1.4.17 drops the symmetry of the bisimulation game: the challenge transitions may only be launched by one of the processes in the pairs. We have seen that simulation equivalence is strictly coarser than bisimilarity. Unfortunately, it does not respect deadlock. In this section we discuss a few refinements of simulation equivalence without this drawback. They are coinductively defined, much like bisimilarity, while being coarser than bisimilarity. Thus they can allow us to use coinduction in situations where bisimilarity may be over-discriminating. Another possible advantage of a simulationlike relation is that it naturally yields a preorder (with all the advantages mentioned in Section 5.5). With respect to simulation-based relations, however, bisimilarity remains mathematically more robust and natural. The most interesting refinements we examine are represented by ready similarity and coupled similarity.

We begin in Section 6.1 with complete simulation, and continue in Section 6.2 with ready simulation. They are to simulation what complete trace equivalence and failure (or ready) equivalence are to trace equivalence. In Section 6.3 we discuss two-nested simulation equivalence. In Section 6.4 we consider the weak versions of the relations in the previous sections. In Section 6.5 we present coupled similarity, which aims at relaxing the bisimulation requirements on the internal actions of processes. Finally, in Section 6.6, we summarise the various equivalences discussed in this and previous chapters.

Complete simulation

The arguments about the deadlock-insensitivity of trace equivalence in Section 1.3.2, such as the equality between the processes in Figure 1.4, apply to simulation equivalence too.

In the first part of this chapter we present (yet) another characterisation of bisimilarity, namely bisimilarity as a testing equivalence. In a testing scenario two processes are equivalent if no experiment can distinguish them. An experiment on a process is set up by defining a test, that is, intuitively, a pattern of demands on the process (e.g., the ability of performing a certain sequence of actions, or the inability of performing certain actions). Depending on how the process behaves when such a test is conducted, the observer emits a verdict about the success or failure of the experiment. The experiments are means of understanding how the processes react to stimuli from the environment. A testing scenario has two important parameters.

1. How are observations about the behaviour of a process gathered? In other words, what can the observer conducting the experiment do on a process? For instance, is he/she allowed to observe the inability of a process to perform certain actions? Is he/she allowed to observe whether a process can immediately perform two distinct actions? Is he/she allowed to make a copy of a process? These decisions are embodied in the language for the tests.

2. How is the success or the failure of an experiment determined?

The choice for these parameters has an impact on the distinctions that can be made on the processes. We will set up a testing scenario whose induced equivalence is precisely bisimilarity.

In this chapter we introduce some common process operators, which impose a structure on processes and bring in concepts from algebra. The operators we consider are inspired by those of the Calculus of Communicating Systems (CCS) [Mil89, AILS07], one of the most studied process calculi. Given a process calculus (or a language), one obtains an LTS by providing, for each operator, a set of inference rules that determine the possible transitions of the processes in the syntax-driven fashion of Plotkin's Structured Operational Semantics (SOS) [Plo04a, Plo04b]. We restrain from going into a thorough discussion on process calculi. We do present, however, nearly all the operators of CCS. What here, and in the following chapters, is called CCS is in fact very close to the standard presentation as in, e.g., [Mil89, AILS07]. The only technical differences are that we do not introduce the relabelling operator, and, for writing processes with an infinite behaviour, we use constant symbols instead of recursive process equations; each constant comes with its own transitions. These differences are discussed in Remark 3.5.8 and following exercises, and Remark 3.2.3.

The chapter offers a number of examples of the bisimulation proof method. The method is used to prove that bisimilarity is preserved by the operators considered (and that therefore it is a congruence in CCS), to prove some basic algebraic laws, and in various examples.