Efficient Computation of LALR(1) Look-Ahead Sets

Abstract

Two relations that capture the essential structure of the problem of computing LALR(1) look-ahead sets are defined, and an efficient algorithm is presented to compute the sets in time linear in the size of the relations. In particular, for a PASCAL grammar, the algorithm performs fewer than 15 percent of the set unions performed by the popular compiler-compiler YACC.When a grammar is not LALR(1), the relations, represented explicitly, provide for printing useroriented error messages that specifically indicate how the look-ahead problem arose. In addition, certain loops in the digraphs induced by these relations indicate that the grammar is not LR(k) for any k.Finally, an oft-discovered and used but incorrect look-ahead set algorithm is similarly based on two other relations defined for the fwst time here. The formal presentation of this algorithm should help prevent its rediscovery.

A general model for fixed look-ahead lr parsers

“…DeRemer and Pennello (see [9]) suggest that the latter class is a subset of the former, at least for the case k= m = 1. We prove here that no relationship exists.…”

“…For example, consider a production A+wAy, where A and o generate the empty string. In this case, if an LR(0) state s with a transition on A appears on the top of a stack in a look-ahead item, Closure would add items forever as follows: This situation is equivalent to that posed by DeRemer and Pennello [9]. In computing LALR(1) look-ahead sets, they define a relation among nonterminal transitions in the LR(0) parser called reads:…”

“…NQLALR(k,m) is a generalization of NQLALR(l), an oversimplification of LALR(1) described by DeRemer and Pennello [ 9 ] . NQLALR(1) is a particular case of our NQLALR(k,m) technique: it is identical to NQLALR(1,l).…”

“…The differences among them reside in the way the look-ahead sets are computed: one of the best LALR(1) algorithms [9] propagates (and collects) the look-ahead symbols throughout the LR(0) parser according to the graph induced by a relation among nonterminal transitions in the LR(0) automaton. The NQLALR(1) algorithm [9] is similar, except that the relation is among states in the LR(0) parser, rather than among nonterminal transitions. SLR(1) lookahead sets [8] are (the traditional) Follow sets for certain nonterminals, which can be computed directly from the grammar.…”

“…4) The relationship among these classes of context-free grammars have been misunderstood in the past. Specifically, DeRemer and Pennello [9] claim that SLR(l)cNQLALR(l). We prove here that this is not true, and generalize the result to multisymbol look-ahead.…”