2011; Roark, Sproat, and Shafran 2011) w 1 , w 2 over the tropical weights w 1 and w 2 with the operations ⊕ and ⊗: The PDA algorithms described in Section 3 are valid under this new semiring because it is commutative and has the path property.
the PDA representing {s} • G is constructed so that the translation grammar score appears in both w 1 and w 2 (i.e., it is duplicated).