Generality Considered Harmful— A Critique of Descriptive Semantics∗

We would like in this note to offer a criticism of the use of mathematics in most of the current work in the semantics of programming languages. Our complaint is not the wrong amount of mathematics (either too little or too much) is being used, or that the wrong kind is being used, but rather than mathematics is being used in the wrong way. In semantics, and computer science as a whole, there have always been two points of view concerning the rôle of mathematics. One point of view sees mathematics as playing primarily a passive rôle. According to this point of view the entities considered by computer scientists (machines, language, systems) are shaped by forces outside mathematicians’ control; the job of the mathematician is therefore to develop the necessary tools to study computer science, i.e. to describe, to model, to classify. This we might call the “descriptive” approach. The other point of view sees mathematics as playing primarily an active rôle. According to this point of view, machines, languages, and systems are (or should be) our own creations, and we can freely choose to create them so that they conform to mathematically simple principles. The mathematical tools developed are used mainly for design rather than study. Mathematics is used not so much to describe existing objects as to plan new objects. This we shall call the “prescriptive” approach. Our criticism of semantics, be it operational or mathematical, Vienna Definition Language [5] or Scott–Strachey semantics [6], functional or relational, is that it almost always takes the descriptive rather than the prescriptive approach. In the field of syntax, the difference between the descriptive and prescriptive approaches, and the superioriy of the latter, can clearly be seen. The first programming languages (such as FORTRAN and COBOL) were designed in an incremental and ad hoc manner and grammar (as well as everything else) was specified by an informal mixture of English and examples. This was quickly seen to be inadequate, and, in the course of designing Algol 58, Backus, Naur and others (and Chomsky independently) devised what are now known as context-free grammars (or BNF). These grammars are simple and powerful, but, even so, are still not adequate to describe completely the peculiarities of FORTRAN. The Algol 58 committee recognized the limitations of BNF but did not try to extend it to make it more general; instead, they fashioned Algol 60 in such a way as to allow a simple BNF specification. This is an example of the prescriptive approach. Since that time however, a number of researchers have been unable to resist the temptation to “rectify” the descriptive inadequacies of BNF. Ths most comprehensive of these generalizations is the system of two-level grammars of Van Wijngaarden, invented to describe Algol 68 [8]. These two-level grammars are completely general in the sense that any recursively enumerable set (for example, the set of valid first-order formulas) can be so specified. Yet in spite of this generality it cannot be said that two-level grammars are really succesful. In fact, we maintain that their lack of success is a consequence of their generality. These grammars offer no guidance to the language designer. True, the grammars can describe any language, but in general the description will be so complex and unnatural as to be almost useless. In particular, there is no general way to devise a parsing algorithm for a grammar. This is not surprising, since there may be no such algorithm at all (of any kind) that can recognize the language.