Describing the syntax of programming languages using conjunctive and Boolean grammars

TL;DR

This paper demonstrates how conjunctive and Boolean grammars can precisely describe the syntax of a programming language, enabling feasible parsing algorithms that surpass traditional context-free grammar limitations.

Contribution

It introduces the first complete Boolean grammar for a programming language's syntax and shows how to transform it into an unambiguous conjunctive grammar with efficient parsing.

Findings

Boolean grammar allows complete syntax specification

Parsing with Boolean grammars can be done in polynomial time

Transformation to conjunctive grammar enables square-time parsing

Abstract

A classical result by Floyd ("On the non-existence of a phrase structure grammar for ALGOL 60", 1962) states that the complete syntax of any sensible programming language cannot be described by the ordinary kind of formal grammars (Chomsky's ``context-free''). This paper uses grammars extended with conjunction and negation operators, known as conjunctive grammars and Boolean grammars, to describe the set of well-formed programs in a simple typeless procedural programming language. A complete Boolean grammar, which defines such concepts as declaration of variables and functions before their use, is constructed and explained. Using the Generalized LR parsing algorithm for Boolean grammars, a program can then be parsed in time $O(n^4)$ in its length, while another known algorithm allows subcubic-time parsing. Next, it is shown how to transform this grammar to an unambiguous conjunctive grammar, with square-time parsing. This becomes apparently the first specification of the syntax of a programming language entirely by a computationally feasible formal grammar.