Say we have some BNF rules for example:
$$A ::= B C$$ $$Dx ::= ExFx$$ $$x ::= y$$
So we might start with $Dx$ and apply rule 2 to get $Ex Fx$ and then apply rule 3 to get $Ey Fy$. But starting with $Dy$ we can't get anything. Unless...
What if we extend the BNF rules to let the rules apply to other rules. Then we could take rule 3 and apply it to rule 2 (interpreted as a string) to get a new rule:
$$Dy ::= EyFy$$
My question is, if we allow such creation of new rules like this. What kind of grammar do we end up with? What is the name of it? Is it equivalent to church lambda calculus?
Using these rules we could express the grammar where the quantity matches the noun and determiner like this:
$$amount ::= singular|plural \\ NP(amount) ::= det(amount)\ \ noun(amount) \\ det(singular) ::= a | one \\ det(plural) ::= some | two | three \\ noun(singular) ::= cat | dog \\ noun(plural) ::= cats| dogs $$
From which we would get $some\ cats$ and $one\ dog$ for example.