LR(1) Parsing

1. Simply using Follow sets to interpret look ahead symbols may give less information than is really available.
   • Consider the grammar:

     E ( L , E )
     E S
     L L , E
     L E
     S ident
     S ( S )

     The state reached on input `( S' contains an SLR(1) conflict but 1 symbol look ahead is enough to allow us to parse.

     To see why, build the LR(0) machine. The conflict is between the items [ S ( S . ) ] and [ E S . ] and ")" is clearly in the Follow set of E. However, if one reduces using the production in the reduce item, one would quickly ends up in a state where you further reduce the E to an L. Then, you end up in a state where the only possible action is to shift in a comma. So, if the next input is not a comma, reducing by E S is a dead end.

2. LR(1) parsing is a rather straightforward generalization of LR(0) parsing that keeps track of both what points in what productions we might be up to and what might follow the rhs's of the productions we are working on.
3. We start with plenty of new (but familiar) definitions.

   LR(1) item

   Given a grammar G, we say that [ N ₁ . ₂ , a ] is an LR(1) item or LR(1) configuration for G if N _{1 2} is a production in G and a ( V_t U ). The symbol `a' is called the lookahead.

   Configuration Set

   We will refer to a set of LR(1) items as an LR(1) configuration set.

   Valid item

   Given a grammar G, we say that an LR(1) item [ N ₁ . ₂, a ] is valid for ( V_n U V_t )^* if there is a rightmost derivation

   S N omega _{1 2} omega

   such that ₁ = and a First(omega).