SLR(1) Parsing

1. Given the notion of the "follow" set, we can illustrate the use of look-ahead in LR parsing, by considering the simplest form of look-ahead LR parsing -- SLR(1) parsing (that's S for simple).
2. Recall the grammar

   < S > a < S > b < S >  | 

3. The LR(0) machine for this grammar contains three states with LR(0) conflicts (two of which were shown above):
   • The initial state is composed of the items:

     [ < S' > . < S > $ ]
     [ < S > . a < S > b < S > ]
     [ < S > . ]

   • Starting from the initial state on input "a" we reach the the state:

     [ < S > a . < S > b < S > ]
     [ < S > . a < S > b < S > ]
     [ < S > . ]

   • Starting from the initial state on input "aSb" we reach the the state:

     [ < S > a < S > b . < S > ]
     [ < S > . a < S > b < S > ]
     [ < S > . ]

4. For this grammar, a is not in Follow(S). So, we should never reduce using the production < S > if the next "input" symbol is a. Therefore, the three states with LR(0) conflicts really don't have conflicts at all.
5. In general, we will say that a set of LR(0) items contains an SLR(1) conflict if either:
   1. It contains two reduce items [ N ₁ . ] and [ M ₂ . ] such that the intersection of Follow(N) and Follow(M) is non-empty, or
   2. It contains a reduce item [ N ₁ . ] and a shift item [ M ₂ . x ₂ ] such that x Follow(N).
6. If the LR(0) machine for a grammar G contains no states with SLR(1) conflicts we say that G is an SLR(1) grammar.
7. An SLR(1) parser for an SLR(1) grammar G behaves as follows in state when the next input symbol is "x":
   • reduce using production N if [ N . ] , and x Follow(N).
   • shift in next input if contains one or more items of the form [ N . x ].
   • error otherwise.