Abstract
The famous BNF grammar notation, as introduced and used in the Algol 60 report, was subsequently followed by numerous notational variants (EBNF, ABNF, RBNF, etc.), and later by a new formal “grammars” metalanguage used for discussing structured objects in Computer Science and Mathematical Logic. We refer to this latter offspring of BNF as MBNF (Math-BNF), and to aspects common to MBNF, BNF, and its notational variants as BNF-style. MBNF is sometimes called “abstract syntax”, but we avoid that name because MBNF is in fact a concrete form and there is a more abstract form. What all BNF-style notations share is the use of production rules like (P) below which state that “every instance of \(\circ _i\) for \(i \in \{1,...,n\}\) is also an instance of \(\bullet \)”.
However, MBNF is distinct from all variants of BNF in the entities and operations it allows. Instead of strings, MBNF builds arrangements of symbols that we call math-text and allows “syntax” to be built by interleaving MBNF production rules and other mathematical definitions that can contain chunks of math-text. The differences between BNF (or its variant forms) and MBNF have not been clearly presented before. (There is also no clear definition of MBNF anywhere but this is beyond the scope of this paper.)
This paper reviews BNF and some of its variant forms as well as MBNF, highlighting the differences between BNF (including its variant forms) and MBNF. We show via a series of detailed examples that MBNF, while superficially similar to BNF, differs substantially from BNF and its variants in how it is written, the operations it allows, and the sets of entities it defines. We also argue that the entities MBNF handles may extend far outside the scope of rewriting relations on strings and syntax trees derived from such rewriting sequences, which are often used to express the meaning of BNF and its notational variants.
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Notes
- 1.
For example, consider an abstract syntax tree (AST) representing \(\lambda x.e\). An AST is a tree where each branch goes to a syntactic evaluation of a metavariable and each node is either a metavariable assignment which contains no further evaluations or a function taking metavariables, which represents an evaluation. In an AST for \(\lambda x.e\), we would not be interested that the x and the e are arranged with a dot between them and a \(\lambda \) in front of them. Rather, would just be a name for aparticular function taking two arguments of an appropriate type.
- 2.
We chose ESOP 2012, but we could equally pick any other conferences. Because the first book we picked contained an abundance of challenging instances of MBNF, our wider searching has mainly been to find even more challenging examples. We will be happy to receive pointers to additional interesting cases. We also checked the POPL 2017 proceedings [11] and found that out of 46 papers using BNF-style notation, not one used notation exactly corresponding to the EBNF [20], ABNF [6] or RBNF [10] standards and only one [16] could possibly be thought of as EBNF or ABNF with variant syntax. Out of the other 45 POPL 2017 papers featuring BNF-style notation, 44 use what we call MBNF.
- 3.
Here a metavariable is a variable at the meta-level which denotes something at an object-level.
- 4.
- 5.
The root node is on the spine. If A is applied to B by an application on the spine, the root node of A is on the spine and the root node of B is not. If a node on the spine is an abstraction each of its children is on the spine (i.e., every node appearing on the furthest left hand side of the tree is on the spine).
- 6.
A balanced segment is one where each application has a matching abstraction and where each application/abstraction pair contains a balanced segment.
- 7.
Actually a slightly weaker condition than the one we give here is probably sufficient for the Barendregt convention to hold, but it would be more complicated to state.
- 8.
As previously noted we avoid that name, because MBNF is a concrete form.
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Quinlan, D., Wells, J.B., Kamareddine, F. (2019). BNF-Style Notation as It Is Actually Used. In: Kaliszyk, C., Brady, E., Kohlhase, A., Sacerdoti Coen, C. (eds) Intelligent Computer Mathematics. CICM 2019. Lecture Notes in Computer Science(), vol 11617. Springer, Cham. https://doi.org/10.1007/978-3-030-23250-4_13
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