Abstract: A source sentence is decoded in an iterative manner. At each step a set of partially constructed target sentences are collated, each of which has a score or an associated probability, computed from a language model score and a translation model score. At each iteration, a family of exponentially many alignments is constructed and the optimal translation for this family is found out. To construct the alignment family, a set of transformation operators is employed. The described decoding algorithm is based on the Alternating Optimization framework and employs dynamic programming. Pruning and caching techniques may be used to speed up the decoding.