In all cases, we prune the lattices/hypergraphs to a density of 30 using forward-backward pruning (Sixtus and Ortmanns, 1999).
because the right-hand side of r e has n nonterminals, the arity of e is |e| = n. Let T (e) = {v 1 , ..., v n } denote the tail nodes of e. We now assume that each tail node v i ∈ T (e) is associated with the upper envelope over all candidate translations that are induced by derivations of the corresponding nonterminal symbol X i .