On the Complexity of Alignment Problems in Two Synchronous Grammar Formalisms

The alignment problem for synchronous grammars in its unrestricted form, i.e. whether for a grammar and a string pair the grammar induces an alignment of the two strings, reduces to the universal recognition problem, but restrictions may be imposed on the alignment sought, e.g. alignments may be 1 : 1, island-free or sure-possible sorted. The complexities of 15 restricted alignment problems in two very different synchronous grammar formalisms of syntax-based machine translation, inversion transduction grammars (ITGs) (Wu, 1997) and a restricted form of range concatenation grammars ((2,2)-BRCGs) (Søgaard, 2008), are investigated. The universal recognition problems, and therefore also the unrestricted alignment problems, of both formalisms can be solved in time O(n6|G|). The complexities of the restricted alignment problems differ significantly, however.