Recognition is not parsing - SPPF-style parsing from cubic recognisers

In their recogniser forms, the Earley and RIGLR algorithms for testing whether a string can be derived from a grammar are worst-case cubic on general context free grammars (CFG). Earley gave an outline of a method for turning his recognisers into parsers, but it turns out that this method is incorrect. Tomita’s GLR parser returns a shared packed parse forest (SPPF) representation of all derivations of a given string from a given CFG but is worst-case unbounded polynomial order. The parser version of the RIGLR algorithm constructs Tomita-style SPPFs and thus is also worst-case unbounded polynomial order. We have given a modified worstcase cubic GLR algorithm, that, for any string and any CFG, returns a binarised SPPF representation of all possible derivations of a given string. In this paper we apply similar techniques to develop worst-case cubic Earley and RIGLR parsing algorithms.