The Correctness of LR(0) parsing

The Correctness of LR(0) parsing

1. The correctness of the LR(0) parsers we have discussed rests on the theorem:

   Theorem

   [ N ₁ . ₂ ] ( ₀, ) iff [ N ₁ . ₂ ] is valid for .

2. Rather than completely prove the theorem, I would like to prove it in one direction (the easy one) and let you convince yourselves of the other direction. In particular, we will prove that:

   [N -> ₁ . ₂ ] ( ₀, ) only if [ N ₁ . ₂ ] is valid for .

3. We can break the proof of this half of the theorem into two lemmas:

   Lemma 1: Given a set of LR(0) items valid for some (V_n U V_t)^*, all items in closure() are valid for .

   Lemma 2: For kernel items, [N -> ₁ . ₂ ] ( ₀, ) only if [ N ₁ . ₂ ] is valid for .

The Correctness of LR(0) parsing