Formal grammar with variables for consistent substitutions

Question

In a rewriting system, suppose the production rule S→xAyAz (or <S>:=x<A>y<A>z, in BNF), where S and A are nonterminal symbols and x, y, z are any string.

This rule produces the string xa₁ya₂z, where a₁ and a₂ are strings substituted by the nonterminal symbol A. The substitutions a₁ and a₂ may be equal or different strings.

But what if I want to a₁ and a₂ to be the same string; that is, if the nonterminal A be substituted by the same instance in both its occurrences in that production rule?

I used a kind of BNF I created informally for such purpose, with <v:A> instead of <A>: a kind of "typed variable", whose type is a nonterminal symbol and the variable is the same substitution for all its occurrences. Thus, <S>:=x<v:A>y<v:A>z, and <v:A> is substituted for the same string in its both occurrences.

What is the name of such a grammar that uses variables for consistent substitution of a nonterminal? And there is a grammar formalism for it?

Answer 1

You've created an affix grammar, so named because they were created to deal with affixes (prefixes and suffixes) in natural languages.

Specifically, this is a type of affix grammar where the "affixes" (what you call typed variables) are defined by the same type of grammar rather than coming from a simpler (finite) grammar. This creates a two-level grammar, specifically a type known as a Van Wijngaarden grammar after its creator.

It turns out that this type of grammar is in fact Turing-complete, so non-trivial semantic properties of such grammars are undecidable by Rice's theorem. On the one hand this makes them extremely powerful; on the other hand, it makes them not as useful in practice, since you could use some other Turing-complete system instead.