Completeness Conditions for Mixed Strategy Bidirectional Parsing

It has been suggested that, in certain circumstances, it might be useful for a grammar writer to annotate which rules are to be used bottom-up and which are to be used top-down within a parser, using a bidirectional variant of the active chart parsing technique. The formal properties of such systems have not been fully explored. One limitation of this mixed strategy technique is that certain annotations of rules can lead to incompleteness; that is, there may be valid analyses of the input string that cannot be found by the parser. We formalize a fairly natural notion of mixed strategy bidirectional parsing for context-free grammars, in which one or more symbols within a rule may be annotated as "triggers," so that the rule is either top-down (triggered from its lefthand side), or bottom-up (triggered from element(s) of its right-hand side). We define a decidable property of annotated grammars, such that any grammar with this property is provably complete. There are, however, some complete annotations of grammars that fall outside this decidable class. We show that membership of this wider class is undecidable. These results suggest that the mixed strategy approach is of rather limited usefulness, regardless of whether it is empirically efficient or not.