Process algebra focuses on the specification and manipulation of process terms as induced by a collection of operator symbols. Process algebras such as CCS [24, 84, 86], CSP [67, 68, 95] and ACP [11, 5, 43] have proven to be nice basic languages for the description of elementary parallel systems and they are well equipped for the study of behavioural properties of distributed systems. Fundamental to process algebra is a parallel operator, to break down systems into their concurrent components. Most process algebras contain basic operators to build nite processes, communication operators to express concurrency, and some notion of recursion to capture innite behaviour. Moreover, a special hiding operator allows one to abstract away from internal computations. In process algebras, each operator in the languageis given meaning through a characterising set of equations called axioms. If two process termscan be equated by means of the axioms, then they constitute equivalent processes. Thus the axioms form an elementary basis for equational reasoning about processes. System behaviour generally consists of a mix of processes and data. Processes are the control mechanisms for the manipulation of data. While processes are dynamic and active, data are static and passive. In algebraic specication, each data type is dened by declaring a collection of function symbols, from which one can build data terms, together with a set of axioms, saying which data terms are equal. Algebraic specication allows one to give relatively simple and precise denitions of abstract data types. A major advantage of this approach is that it is easily explained and formally dened, and that it constitutes a uniform framework for dening general data types. Moreover, all properties of a data type must be explicitly denoted, and henceforth it is clear which assumptions can be used when proving properties about data or processes. Term rewriting [4] provides a straightforward method for implementing algebraic specications of abstract data types. Concluding, as long as one is interested in clear and precise specications, and not in optimised implementations, algebraic specication is the best available method. However, one should be aware that it does not allow one to use high-level constructs for compact specication of complex data types, nor optimisations supporting fast computation (such as decimal representations of natural numbers). Process algebras tend to lack the ability to handle data. In case data becomes part of a process theory, one often has to resort to innite sets of axioms where variables are indexed with data values. In order to make data a rst class citizen in the study of processes, the language micro CRL[59] has been developed,1 denoted CRL (or mCRL, if Greek letters are unavailable). Basically, CRL is based on the process algebra ACP, extended with equational abstract data types. In order to intertwine processes with data, actions and recursion variables can be parametrised with data types.

An overview of the book

This text is set up as follows. Chapter 2 gives an introduction into the algebraic specication of abstract data types. Chapter 3 provides an overview of process algebra, and explains the basics of the specication language CRL. In Chapter 5 it is explained how one can abstract away from the internal computation steps of a process. Chapter 6 contains a number of CRL specications of protocols from the literature, together with extensive explanations to guide the reader through these specications. In Chapter 8 a number of standard process algebraic techniques are described that can be used in the verication of CRL specications. In Chapter 10, these techniques are applied in the verication of the tree identify protocol and a sliding window protocol. Chapter 7 describes algorithms on graphs. In Chapter 9, techniques are presented to analyse and adapt CRL specications on a symbolic level. Chapter 11 gives an overview and some applications of an extension of the language CRL with time. Finally, Chapter 12 contains a short overview of the CRL toolset.

Preface

Part I. A Classic Theory of Reactive Systems: 1. Introduction; 2. The language CCS; 3. Behavioural equivalences; 4. Theory of fixed points and bisimulation equivalence; 5. Hennessy-Milner logic; 6. Hennessy-Milner logic with recursive definitions; 7. Modelling and analysis of mutual exclusion algorithms; Part II. A Theory of Real-Time Systems: 8. Introduction; 9. CCS with time delays; 10. Timed automata; 11. Timed behavioural equivalences; 12. Hennessy-Milner logic with time; 13. Modelling and analysis of Fischer’s algorithm; Appendix; Bibliography; Index.