What Does This Notation Mean Anyway?

TL;DR

This paper explores the ambiguous notation of Math-BNF (MBNF), illustrating its differences from BNF, and proposes a definition of syntactic math text (SMT) to clarify its usage and structure.

Contribution

It provides examples of MBNF, highlights its differences from BNF, and introduces a formal definition of SMT to improve understanding and application.

Findings

MBNF differs from BNF in entities and operations

A set of examples illustrating MBNF usage

Proposed definition of SMT for better clarity

Abstract

Following the introduction of BNF notation by Backus for the Algol 60 report and subsequent notational variants, a metalanguage involving formal "grammars" has developed for discussing structured objects in Computer Science and Mathematical Logic. We refer to this offspring of BNF as Math-BNF or MBNF, to the original BNF and its notational variants just as BNF, and to aspects common to both as BNF-style. What all BNF-style notations share is the use of production rules roughly of this form: $$\bullet \mathrel{::=} \circ_1 \mid \cdots \mid \circ_n $$ Normally, such a rule says "every instance of $\circ_i$ for $i \in \{1, \ldots, n\}$ is also an in stance of $\bullet$". MBNF is distinct from BNF in the entities and operations it allows. Instead of strings, MBNF builds arrangements of symbols that we call math-text. Sometimes "syntax" is defined by interleaving MBNF production rules and other mathematical definitions that can contain chunks of math-text. There is no clear definition of MBNF. Readers do not have a document which tells them how MBNF is to be read and must learn MBNF through a process of cultural initiation. To the extent that MBNF is defined, it is largely through examples scattered throughout the literature and which require readers to guess the mathematical structures underpinning them. This paper gives MBNF examples illustrating some of the differences between MBNF and BNF. We propose a definition of syntactic math text (SMT) which handles many (but far from all) uses of math-text and MBNF in the wild. We aim to balance the goal of being accessible and not requiring too much prerequisite knowledge with the conflicting goal of providing a rich mathematical structure that already supports many uses and has possibilities to be extended to support more challenging cases.