This book is about formal specification and design techniques, including both algebraic specifications and state-based specifications.

The construction and maintenance of complex software systems is a difficult task and although many software projects are started with great expectations and enthusiasm, it is too often the case that they fail to achieve their goals within the planned time and with the given resources. The software often contains errors; attempts to eliminate the errors give rise to new errors, and so on. Moreover, the extension and adaptation of the software to new tasks turns out to be a difficult and tedious task, which seems unsuitable for scientific methods.

This unsatisfactory situation can be improved by introducing precise specifications of the software and its constituent parts. When a piece of software P has a precise specification S say, then ‘P satisfies S’ is a clear statement that could be verified by reasoning or that could be falsified by testing; users of P can read S and rely on it and the designer of P has a clearly formulated task. When no precise specifications are available, there are hardly any clear statements at all, for what could one say: ‘it works’ or more often ‘it almost works’? Without precise specifications, it becomes very difficult to analyse the consequences of modifying P into P', for example, and to make any clear statements about that modification. Therefore it is worthwhile during the software development process to invest in constructing precise specifications of well-chosen parts of the software system under construction. Writing precise specifications turns out to be a considerable task itself.

Introduction

The conception, construction, maintenance and usage of computer-based systems are difficult tasks requiring special care, skills, methods and tools. Program correctness is a serious issue and in addition to that, the size of the programs gives rise to problems of complexity management. Computers are powerful machines which can execute millions of instructions per second and manipulate millions of memory cells. The freedom offered by the machine to its programmer is large; often it is too large, in the sense that the machine does not enforce order and structure upon the programs. Computer-based systems are artificial systems and therefore there are no natural system partitionings and interface definitions. All structure is man-made and all interfaces must be agreed upon and communicated to all parties involved. The description and communication of system structures and interfaces turns out to be a non-trivial task and ‘specification languages’ have become an active area of research and development in computer science. When discussing ‘language’ we must distinguish explicitly between syntactic objects and semantic objects. Wittgenstein has expressed this idea as follows:

i.e. the proposition represents the existence or non-existence of certain states of affairs. The propositions are syntactic objects and in this text we shall call them specifications. To describe a state of affairs concerning the natural world and concerning human interaction, natural language is the tool par excellence; to describe a state of affairs concerning computer-based systems, special languages are required in addition to that. The situation is typical: special restricted domains require special languages and this is also the case for the domain of computer-based systems.

Introduction

All the previous chapters are about techniques for unambiguously specifying hardware/software systems and transforming abstract specifications into efficient programs. One important motivation for presenting these techniques is the fact that it is often useful to have a distinction between the external view and the internal view of a system. The external view can take the form of a formal specification and it can be optimised with respect to abstractness, compactness and clarity. The internal view is a program which is devised with efficiency in mind. To have two descriptions corresponding to these two views can be considered as a separation of concerns: it helps to manage the complexity of large systems.

This approach introduces additional formal texts when compared with the older approaches dealing mostly with programs. As a consequence, care is needed to maintain the overview of all formal texts that arise when designing large systems.

This chapter presents two techniques developed in the context of COLD-K which serve for keeping this overview. These are certainly not the only useful techniques; they should be complemented with additional graphical techniques and classical software engineering techniques for configuration management, project management, etc. The first technique is to use simple pictures showing the modular structure of a formal specification. This is the topic of Section 11.2. The second technique is to add structure, putting specifications and implementations together in simple language constructs called components and designs. This is the topic of Section 11.3. Finally, Sections 11.4 and 11.5 present a number of applications as well as some concluding remarks.

This monograph promotes specification and programming on the basis of Horn logic with equality. As was pointed out in [Pad88a], this theoretical background equips us with a number of deductive methods for reasoning about specifications and designing correct programs. The term declarative programming stands for the combination of functional (or applicative) and relational (or logic) programming. This does not rule out the design of imperative programs with conditionals, loops and sequences of variable assignments, since all these features have functional or relational equivalents. In particular, variables become “output parameters” of functions. Hence the static view of declarative programming is not really a restriction. Only if correctness conditions concerned with liveness or synchronization are demanded, transition relations must be specified for fixing the dynamics of program execution (cf. Sect. 6.6).

Design specifications

With regard to the overall software design process, the methods considered here are tailored to design specifications, each consisting of a many-sorted signature SIG denoting data, functions and predicates to be specified and a set of Horn clauses over SIG, allowing more or less abstract presentations of declarative programs and the data structures they use and manipulate (cf. Sects. 1.1 and 1.2). Associated with a design specification DS is a requirement specification, the conjecture section of DS, which consists of correctness conditions on DS. In contrast to design axioms, Horn clauses are not always sufficient for specifying requirements. Hence we admit positive Gentzen clauses, which may involve disjunctions and existential quantifiers, in a requirement specification (cf. Sect. 1.4).

Kinds of axioms

In Chapter 2 a number of techniques for setting up algebraic specifications were given. Now we want to do the same for state-based specifications. Recall that a state-based specification serves to describe a system whose model is a ‘class’. A class is a kind of state-machine where each state has an algebra associated with it. Because each such algebra can be viewed as a static world model, classes with their state transitions can be viewed to model dynamic systems. Special language constructs to specify these systems were introduced, such as procedures and the operators of dynamic logic. It is typical for these language constructs that they leave the sort of states implicit.

Therefore the techniques of Chapter 2 are mostly useful for describing the static aspects of states, whereas we need complementary guidelines for dealing with the dynamic aspects of a system. This is the main topic of this chapter. Key notions are that of precondition, postcondition and invariant.

In practice it turns out that there are certain patterns that occur quite often in state-based specifications, providing methodological guidelines for systematically setting up state-based specifications. In this chapter we shall do this, focusing on the axioms, and we adopt a classification distinguishing four distinct roles an axiom can play. These roles correspond to typical syntactic patterns. We shall refer to these by saying that there are four kinds of axioms:

1. properties of all states;

2. invariance properties;

3. properties of the initial state;

4. properties of state transitions including pre- and postcondition axioms and termination axioms.