Two Families of Languages Related to ALGOL

Abstract

A serious drawback in the application of modern data processing systems is the cost and time consumed in programming these complexes. The user's problems and their solutions are described in a natural language such as English. To utilize the services of a data processor, it is necessary to convert this language description into machine language, to wit, program steps. Recently, attempts have arisen to bridge the gap between these two languages. The method has been to construct languages (called problem oriented languages, or that are (i) rich enough to allow a description of a set of problems and their solutions; (ii) reasonably close to the user's ordinary language of description and solution; and (iii) formal enough to permit a mechanical translation into machine language. COBOL and ALGOL are two examples of POL. The purpose of this investigation is to gain some insight into the syntax of POL, in particular ALGOL [1]. Specifically, the method of defining constituent parts of ALGOL 60 is abstracted, this giving rise to a family of sets of strings; and mathematical facts about the resulting family deduced. Now an ALGoL-like definable language (we hesitate to use the inclusive term POL) may be viewed either as one of these sets (the set of sentences) ; or else, as a finite collection of these sets, one of which is the set of sentences, and the remaining, the constituent parts of the language used to construct the sentences. This is in line with one current view of natural languages [4, 5, 6]. The defining scheme for ALOOL turns out to be equivalent to one of the several schemes described by Chomsky [6] in his attempt to analyze the syntax of natural languages. Of course, POL, as special kinds of languages, should fit into a general theory of language. However, it is reasonable to expect that POL, as artificial languages contrived so as to be capable of being mechanically translated into machine language, should have a syntax simpler than that of the natural languages. The technical results achieved in this paper are as follows. Two families of sets (of strings), the family of definable sets and the family of sequentially definable sets, are described. Definable sets are obtained from a system of simultaneous equations, all the equations being of a certain form. This system, essentially parallel in nature, is an abstraction of the ALGOL method of description. Definable sets turn out to be identical to the type 2 languages (with identity) introduced by Chomsky [6]. Sequentially definable sets are obtained from a system