Methods for learning with ambiguous labelings have previously been proposed in the context of multi-class classification (Jin and Ghahramani, 2002), sequence-labeling (Dredze et al., 2009), log-linear LFG parsing (Riezler et al., 2002), as well as for discriminative reranking of generative constituency parsers (Charniak and Johnson, 2005).
figure 3 outlines an example of how (and why) AAST works.