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Head-Corner (HC) parsing has come up in computational linguistics a few years ago, motivated by linguistic arguments. This idea is a heuristic, rather than a fail-safe principle, hence it is relevant indeed to consider the worst-case behaviour of the HC parser. We define a novel predictive head-corner chart parser of cubic time complexity.We start with a left-corner (LC) chart parser, which is easier to understand.Subsequently, the LC chart parser is generalized to an HC chart parser.It is briefly sketched how the parser can be enhanced with feature structures.