A Formal Characterization of Parsing Word Alignments by Synchronous Grammars with Empirical Evidence to the ITG Hypothesis

Deciding whether a synchronous grammar formalism generates a given word alignment (the alignment coverage problem) depends on finding an adequate instance grammar and then using it to parse the word alignment.
But what does it mean to parse a word alignment by a synchronous grammar?
This is formally undefined until we define an unambiguous mapping between grammatical derivations and word-level alignments. This paper proposes an initial, formal characterization of alignment coverage as intersecting two partially ordered sets (graphs) of translation equivalence units, one derived by a grammar instance and another defined by the word alignment. As a first sanity check, we report extensive coverage results for ITG on automatic and manual alignments. Even for the ITG formalism, our formal characterization makes explicit many algorithmic choices often left underspecified in earlier work.