Introduction

Chapter 3 has introduced the notion of transition systems both as an abstract operational model of reactive systems and as a means to give operational semantics to equational theories. The latter has been illustrated by giving an operational interpretation of the equational theory of natural numbers. The aim of this book, however, is to develop equational theories for reasoning about reactive systems. This chapter provides the first simple examples of such theories. Not surprisingly, the semantics of these theories is defined in terms of the operational framework of the previous chapter. For clarity, equational theories tailored towards reasoning about reactive systems are referred to as process theories. The objects being described by a process theory are referred to as processes. The next two sections introduce a minimal process theory with its semantics. This minimal theory is mainly illustrative from a conceptual point of view. Sections 4.4 and 4.5 provide some elementary extensions of the minimal theory, to illustrate the issues involved in extending process theories. Incremental development of process theories is crucial in tailoring a process theory to the specifics of a given design or analysis problem. The resulting framework is flexible and it allows designers to include precisely those aspects in a process theory that are relevant and useful for the problem at hand. Incremental design also simplifies many of the proofs needed for the development of a rich process theory.