This paper studies two methods for training hierarchical MT rules independently of word alignments. Bilingual chart parsing and EM algorithm are used to train bi-text correspondences. The first method, rule arithmetic, constructs new rules as combinations of existing and reliable rules used in the bilingual chart, significantly improving the translation accuracy on the German-English and Farsi-English translation task. The second method is proposed to construct additional rules directly from the chart using inside and outside probabilities to determine the span of the rule and its non-terminals. The paper also presents evidence that the rule arithmetic can recover from alignment errors, and that it can learn rules that are difficult to learn from bilingual alignments.