Is there any official terminology about something like double quotes “” grammar?

Question

In many programming language string is a token.

For example:

 token               ::= '"' string
                       | digit nat

 string              ::= char string
                       | '"'

 nat                 ::= digit nat
                       | ϵ

This is a LL(1) grammar for some programming language's toke grammar.

When parsing a string, there is no need to check follow set, because there is a " at the end of each string.

Comparing with nat, string is more easy to parse.

My question is

Is there any official terminology about this kind of grammar?

Thanks.

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Editing:

There was some mistake in the original grammar, thanks @rici for pointing out my mistakes.

Answer 1

FOLLLOW sets are not used in the parsing of either string or nat. In both cases, the parser merely needs to determine if the input is in a set of valid symbols. In the case of nat, the valid symbols are digits; in the case of string, they are characters other than ". (In real languages, the parser would also be checking for \). But in both cases, a check is necessary, and there's no good criterion for saying one test is "simpler" than another. (In practice, both checks are likely to be a simple table lookup. So they're O(1).)

FOLLOW sets are only needed when the grammar contains ε productions. Even then, the parser's actions are not complicated. What is more complicated is building the parser, something which only happens once. It's not really that big of a deal, but it's sufficiently notable that "ε-free grammars" are a thing. I don't think there's any common vocabulary to describe the difference between explicitly and implicitly-terminated repetition, and any way the distinction will be very hard to define strictly. Your string would be parsed by a parser which used a different rule to collect the trailing ", and it's quite possible that the two parsers end up with the same implementation.