We thank all four reviewers for their thoughtful reviews, and are happy that they value the contribution of a new

's suggestion to draw the external nodes on the left-hand side differs from But this wouldn't be as flexible as we'd like; for example, we'd like to query a HMM for the But conjunction is an operation on FGGs, not factor graphs, so at the time of conjunction, no renaming has taken place. We agree that the notation should be improved and will think about how to do so. Lemma 15 does not change the generated graphs and cannot change their treewidth. We mean that an FGG can't generate the