Context-free grammars, their implications, and their uses have been understood and well studied for more than a half-century, but the author demonstrates that there is more to learn and to make explicit by articulating two characteristics of grammars and equations, one new and one old but forgotten, and by presenting a new proof of a familiar characteristic of context-free grammars.

After a concise introduction that deftly summarizes what is to follow, the paper proves an unsurprising but unpublished finding: a solution to a system of equations that corresponds to a grammar is the smallest such solution if it meets a specific criterion. The paper also proves the little-noted finding that if a grammar has more than one solution, it has a nonempty unguarded set. These two results then drive a method for determining the grammar of a language and a proof of the well-known finding that if a language is accepted by a pushdown automaton, it is context free.

Readers interested in grammars and the formal aspects of computer science will find this paper of interest, and writers in these areas will encounter a tightly organized argument that can serve as a model of exposition.