Notes on LALR parse table.

. Merging items may result into conflicts in LALR parsers which did not exist in LR parsers

. New conflicts can not be of shift reduce kind:

- Assume there is a shift reduce conflict in some state of LALR parser with items {[X α .,a],[Y γ .a ß ,b]}

- Then there must have been a state in the LR parser with the same core

- Contradiction; because LR parser did not have conflicts

. LALR parser can have new reduce-reduce conflicts

- Assume states {[X α ., a], [Y ß ., b]} and {[X α ., b], [Y ß ., a]}

- Merging the two states produces {[X α ., a/b], [Y ß ., a/b]}

Merging states will never give rise to shift-reduce conflicts but may give reduce-reduce conflicts and have some grammars which were in canonical LR parser may become ambiguous in LALR parser. To realize this, suppose in the union there is a conflict on lookahead a because there is an item [A α . ,a] calling for a reduction by A α , and there is another item [B ß .a γ ,b] calling for a shift. Then some set of items from which the union was formed has item [A α . ,a], and since the cores of all these states are the same, it must have an item [B ß .a γ ,c] for some c. But then this state has the same shift/reduce conflict on a, and the grammar was not LR(1) as we assumed. Thus the merging of states with common core can never produce a shift/reduce conflict that was not present in one of the original states, because shift actions depend only on the core, not the lookahead. But LALR parser can have reduce-reduce conflicts. Assume states {[X α ., a], [Y ß ., b]} and {[X α ., b], [Y ß ., a]}. Now, merging the two states produces {[X α ., a/b], [Y ß ., a/b]} which generates a reduce-reduce conflict, since reductions by both X α and Y ß are called for on inputs a and b.