Bilexical context-free grammars (2-LCFGs) have proved to be accurate models for statistical natural language parsing. Existing dynamic programming algorithms used to parse sentences under these models have running time of O(∣w∣4), where w is the input string.

A 2-LCFG is splittable if the left arguments of a lexical head are always independent of the right arguments, and vice versa. When a 2-LCFGs is splittable, parsing time can be asymptotically improved to O(∣w∣3). Testing this property is therefore of central interest to parsing efficiency. In this article, however, we show the negative result that splittability of 2-LCFGs is undecidable.

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Author notes


School of Computer Science, University of St. Andrews, North Haugh, St. Andrews, Fife, KY16 9SX, Scotland. E-mail:


Department of Information Engineering, University of Padua, via Gradenigo 6/A, I-35131 Padova, Italy. E-mail: