Semantical BNF

Abstract

BNF (Backus Naur Formalism) grammar is extended by semantic structures. This yields a grammatical formalism that describes semantics uniformly with syntax, semantical BNF. It is formulated in type theoretical grammar, which is a categorial grammar designed for natural languages and formulated in constructive type theory. Semantical BNF applies type theoretical grammar to formal languages, in particular programming languages.

From the point of view of BNF, semantical BNF extends its productions by semantic structures. They allow the grammarian to write rules that generate only semantically as well as syntactically well-formed expressions. This stands in contrast to approaches to grammar in terms of BNF such as attribute grammar. They first generate expressions by BNF productions. Then a separate phase of type checking discards those expressions which are syntactically well-formed but meaningless, i.e. semantically ill-formed. Furthermore, this type checking phase typically does not relate to the syntactic phase of generation in a theoretically rigorous way. Semantical BNF integrates these two phases in terms of a uniform syntactic-semantic grammatical formalism.

From the point of view of constructive type theory, semantical BNF can be seen as its extension with syntactic categories. This allows making syntactic distinctions that are otherwise unavailable in constructive type theory. They are necessary for describing the abstract syntax of a language, and can also be used to vary concrete syntax. A mathematical interpretation function maps the extension back into constructive type theory.

To show how a formal language with variables is described grammatically, Knuth's binary number grammar is interpreted as a semantical BNF grammar, and extended into a small, functional language with typed lambda terms. Semantical BNF is also related to standard contemporary grammars of programming languages that use denotational semantics and to the theory of higher-order abstract syntax.