LFG-DOP (Bod and Kaplan, 1998, 2003) provides an appealing answer to the question of how probabilistic methods can be incorporated into linguistic theory. However, despite its attractions, the standard model of LFG-DOP suffers from serious problems of overgeneration, because (a) it is unable to define fragments of the right level of generality, and (b) it has no way of capturing the effect of anything except simple positive constraints. We show how the model can be extended to overcome these problems. The question of how probabilistic methods should be incorporated into linguistic theory is important from both a practical, grammar engineering, perspective, and from the perspective of ‘pure ’ linguistic theory. From a practical point of view such techniques are essential if a system is to achieve a useful breadth of coverag
