Real-world LR(k > 1) grammars?

Question

Making artificial LR(k) grammars for k > 1 is easy:

Input: A1 B x
Input: A2 B y                   (introduce reduce-reduce conflict for terminal a)
A1   : a
         


        

Answer 1

If a language has an LR(k) grammar, then it has an LR(1) grammar which can be generated mechanically from the LR(k) grammar; furthermore, the original parse tree can be recreated from the LR(1) parse tree. A proof of this theorem appears in Section 6.7 of Parsing Theory Vol. II, Sippu & Soisalon-Soininen; they attribute it to "On the complete covering problem for LR(k) grammars" (1976) by MD Mickunas, in JACM 23:17-30.

So there's no language parseable as LR(k) for k>1 which is not also parseable as LR(1), and consequently there really isn't such as thing as an LR(k) language for values of k other than 0 and 1.

However, there are languages for which an LR(2) grammar is much easier. One classic example is the Posix grammar for yacc, which does not require productions to be terminated with a ;. This is unambiguous, because a production must start with an identifier followed by a colon. Written that way, the grammar is clearly LR(2); by the above theorem, an LR(1) grammar exists but it's nowhere near as clean.